<?xml version="1.0"?>
<ownershipDocument>

    <schemaVersion>X0306</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2020-02-26</periodOfReport>

    <notSubjectToSection16>0</notSubjectToSection16>

    <issuer>
        <issuerCik>0000726854</issuerCik>
        <issuerName>CITY HOLDING CO</issuerName>
        <issuerTradingSymbol>CHCO</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001234855</rptOwnerCik>
            <rptOwnerName>HAGEBOECK CHARLES R</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>25 GATEWATER ROAD</rptOwnerStreet1>
            <rptOwnerStreet2></rptOwnerStreet2>
            <rptOwnerCity>CROSS LANES</rptOwnerCity>
            <rptOwnerState>WV</rptOwnerState>
            <rptOwnerZipCode>25313</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>1</isDirector>
            <isOfficer>1</isOfficer>
            <isTenPercentOwner>0</isTenPercentOwner>
            <isOther>0</isOther>
            <officerTitle>President &amp; CEO</officerTitle>
            <otherText></otherText>
        </reportingOwnerRelationship>
    </reportingOwner>

    <nonDerivativeTable>
        <nonDerivativeHolding>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>50800</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeHolding>
        <nonDerivativeHolding>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>100</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>I</value>
                </directOrIndirectOwnership>
                <natureOfOwnership>
                    <value>by son, Robert B. Hageboeck</value>
                </natureOfOwnership>
            </ownershipNature>
        </nonDerivativeHolding>
        <nonDerivativeHolding>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>100</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>I</value>
                </directOrIndirectOwnership>
                <natureOfOwnership>
                    <value>by daughter, Katherine E. Hageboeck</value>
                </natureOfOwnership>
            </ownershipNature>
        </nonDerivativeHolding>
        <nonDerivativeHolding>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>100</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>I</value>
                </directOrIndirectOwnership>
                <natureOfOwnership>
                    <value>by daughter, Emily F. Hageboeck</value>
                </natureOfOwnership>
            </ownershipNature>
        </nonDerivativeHolding>
        <nonDerivativeHolding>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>5650</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>I</value>
                </directOrIndirectOwnership>
                <natureOfOwnership>
                    <value>by spouse, Samantha D. Hageboeck</value>
                </natureOfOwnership>
            </ownershipNature>
        </nonDerivativeHolding>
        <nonDerivativeHolding>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>1822.8645</value>
                    <footnoteId id="F1"/>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>I</value>
                </directOrIndirectOwnership>
                <natureOfOwnership>
                    <value>by 401(k) Plan and Trust</value>
                </natureOfOwnership>
            </ownershipNature>
        </nonDerivativeHolding>
    </nonDerivativeTable>

    <derivativeTable>
        <derivativeTransaction>
            <securityTitle>
                <value>Restricted Stock Unit</value>
            </securityTitle>
            <conversionOrExercisePrice>
                <value>0.0</value>
                <footnoteId id="F2"/>
            </conversionOrExercisePrice>
            <transactionDate>
                <value>2020-02-26</value>
            </transactionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>A</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>2043</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>0</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>A</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <exerciseDate>
                <footnoteId id="F3"/>
            </exerciseDate>
            <expirationDate>
                <footnoteId id="F3"/>
            </expirationDate>
            <underlyingSecurity>
                <underlyingSecurityTitle>
                    <value>Common Stock</value>
                </underlyingSecurityTitle>
                <underlyingSecurityShares>
                    <value>2043.0</value>
                </underlyingSecurityShares>
            </underlyingSecurity>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>2043</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </derivativeTransaction>
        <derivativeHolding>
            <securityTitle>
                <value>Restricted Stock Unit</value>
            </securityTitle>
            <conversionOrExercisePrice>
                <value>0.0</value>
                <footnoteId id="F2"/>
            </conversionOrExercisePrice>
            <exerciseDate>
                <footnoteId id="F4"/>
            </exerciseDate>
            <expirationDate>
                <footnoteId id="F4"/>
            </expirationDate>
            <underlyingSecurity>
                <underlyingSecurityTitle>
                    <value>Common Stock</value>
                </underlyingSecurityTitle>
                <underlyingSecurityShares>
                    <value>1864.0</value>
                </underlyingSecurityShares>
            </underlyingSecurity>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>1864</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </derivativeHolding>
        <derivativeHolding>
            <securityTitle>
                <value>Restricted Stock Unit</value>
            </securityTitle>
            <conversionOrExercisePrice>
                <value>0.0</value>
                <footnoteId id="F2"/>
            </conversionOrExercisePrice>
            <exerciseDate>
                <footnoteId id="F5"/>
            </exerciseDate>
            <expirationDate>
                <footnoteId id="F5"/>
            </expirationDate>
            <underlyingSecurity>
                <underlyingSecurityTitle>
                    <value>Common Stock</value>
                </underlyingSecurityTitle>
                <underlyingSecurityShares>
                    <value>1749.0</value>
                </underlyingSecurityShares>
            </underlyingSecurity>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>1749</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </derivativeHolding>
        <derivativeHolding>
            <securityTitle>
                <value>Stock Option to Buy</value>
            </securityTitle>
            <conversionOrExercisePrice>
                <value>44.43</value>
            </conversionOrExercisePrice>
            <exerciseDate>
                <footnoteId id="F6"/>
            </exerciseDate>
            <expirationDate>
                <value>2024-03-25</value>
            </expirationDate>
            <underlyingSecurity>
                <underlyingSecurityTitle>
                    <value>Common Stock</value>
                </underlyingSecurityTitle>
                <underlyingSecurityShares>
                    <value>2010.0</value>
                    <footnoteId id="F6"/>
                </underlyingSecurityShares>
            </underlyingSecurity>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>2010</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </derivativeHolding>
        <derivativeHolding>
            <securityTitle>
                <value>Stock Option to Buy</value>
            </securityTitle>
            <conversionOrExercisePrice>
                <value>46.61</value>
            </conversionOrExercisePrice>
            <exerciseDate>
                <footnoteId id="F7"/>
            </exerciseDate>
            <expirationDate>
                <value>2025-02-25</value>
            </expirationDate>
            <underlyingSecurity>
                <underlyingSecurityTitle>
                    <value>Common Stock</value>
                </underlyingSecurityTitle>
                <underlyingSecurityShares>
                    <value>1980.0</value>
                    <footnoteId id="F7"/>
                </underlyingSecurityShares>
            </underlyingSecurity>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>1980</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </derivativeHolding>
        <derivativeHolding>
            <securityTitle>
                <value>Stock Option to Buy</value>
            </securityTitle>
            <conversionOrExercisePrice>
                <value>43.73</value>
            </conversionOrExercisePrice>
            <exerciseDate>
                <footnoteId id="F8"/>
            </exerciseDate>
            <expirationDate>
                <value>2026-02-23</value>
            </expirationDate>
            <underlyingSecurity>
                <underlyingSecurityTitle>
                    <value>Common Stock</value>
                </underlyingSecurityTitle>
                <underlyingSecurityShares>
                    <value>7320.0</value>
                    <footnoteId id="F8"/>
                </underlyingSecurityShares>
            </underlyingSecurity>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>7320</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </derivativeHolding>
        <derivativeHolding>
            <securityTitle>
                <value>Stock Option to Buy</value>
            </securityTitle>
            <conversionOrExercisePrice>
                <value>66.32</value>
            </conversionOrExercisePrice>
            <exerciseDate>
                <footnoteId id="F9"/>
            </exerciseDate>
            <expirationDate>
                <value>2027-02-21</value>
            </expirationDate>
            <underlyingSecurity>
                <underlyingSecurityTitle>
                    <value>Common Stock</value>
                </underlyingSecurityTitle>
                <underlyingSecurityShares>
                    <value>7572.0</value>
                    <footnoteId id="F9"/>
                </underlyingSecurityShares>
            </underlyingSecurity>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>7572</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </derivativeHolding>
    </derivativeTable>

    <footnotes>
        <footnote id="F1">Includes shares acquired pursuant to the Company's 401(k) Plan &amp; Trust during the fiscal year in transactions exempt from 16b under old Rule 16a8(b).  Share totals are reported as of the 12/31/2019 plan valuation date.</footnote>
        <footnote id="F2">Each restricted stock unit represents the right to receive, at settlement, one share of common stock.</footnote>
        <footnote id="F3">One-third of these restricted stock units are scheduled to vest on each of February 26, 2021; February 26, 2022; and February 26, 2023.</footnote>
        <footnote id="F4">One-third of these restricted stock units are scheduled to vest on each of February 27, 2020; February 27, 2021; and February 27, 2022.</footnote>
        <footnote id="F5">One-third of these restricted stock units are scheduled to vest on each of February 28, 2019; February 28, 2020; and February 28, 2021.</footnote>
        <footnote id="F6">Options shares granted from City Holding Company's 2013 Incentive Plan are subject to both time-based and performance-based vesting.  Options shares shall vest as follows: 2,008 shares on each of 3/26/2017 and 3/26/2018 and 2,010 shares on 3/26/2019.</footnote>
        <footnote id="F7">Options shares granted from City Holding Company's 2013 Incentive Plan are subject to both time-based and performance-based vesting.  Options shares will vest as follows: 1,978 shares on 2/26/2019 and 1,980 shares on 2/26/2020.</footnote>
        <footnote id="F8">Options shares granted from City Holding Company's 2013 Incentive Plan are subject to both time-based and performance-based vesting.  Options shares will vest as follows: 3,660 shares on each of 2/24/2019; 2/24/2020; and 2/24/2021.</footnote>
        <footnote id="F9">Options shares granted from City Holding Company's 2013 Incentive Plan are subject to both time-based and performance-based vesting.  Options shares will vest as follows: 2,524 shares on each of 2/22/2020; 2/22/2021; and 2/22/2022.</footnote>
    </footnotes>

    <remarks></remarks>

    <ownerSignature>
        <signatureName>Victoria A. Faw, attorney-in-fact</signatureName>
        <signatureDate>2020-02-28</signatureDate>
    </ownerSignature>
</ownershipDocument>
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X0306
4
2020-02-26
0
0000726854
CITY HOLDING CO
CHCO
0001234855
HAGEBOECK CHARLES R
25 GATEWATER ROAD
CROSS LANES
WV
25313
1
1
0
0
President & CEO
Common Stock
50800
D
Common Stock
100
I
by son, Robert B. Hageboeck
Common Stock
100
I
by daughter, Katherine E. Hageboeck
Common Stock
100
I
by daughter, Emily F. Hageboeck
Common Stock
5650
I
by spouse, Samantha D. Hageboeck
Common Stock
1822.8645
I
by 401(k) Plan and Trust
Restricted Stock Unit
0.0
2020-02-26
4
A
0
2043
0
A
Common Stock
2043.0
2043
D
Restricted Stock Unit
0.0
Common Stock
1864.0
1864
D
Restricted Stock Unit
0.0
Common Stock
1749.0
1749
D
Stock Option to Buy
44.43
2024-03-25
Common Stock
2010.0
2010
D
Stock Option to Buy
46.61
2025-02-25
Common Stock
1980.0
1980
D
Stock Option to Buy
43.73
2026-02-23
Common Stock
7320.0
7320
D
Stock Option to Buy
66.32
2027-02-21
Common Stock
7572.0
7572
D
Includes shares acquired pursuant to the Company's 401(k) Plan & Trust during the fiscal year in transactions exempt from 16b under old Rule 16a8(b).  Share totals are reported as of the 12/31/2019 plan valuation date.
Each restricted stock unit represents the right to receive, at settlement, one share of common stock.
One-third of these restricted stock units are scheduled to vest on each of February 26, 2021; February 26, 2022; and February 26, 2023.
One-third of these restricted stock units are scheduled to vest on each of February 27, 2020; February 27, 2021; and February 27, 2022.
One-third of these restricted stock units are scheduled to vest on each of February 28, 2019; February 28, 2020; and February 28, 2021.
Options shares granted from City Holding Company's 2013 Incentive Plan are subject to both time-based and performance-based vesting.  Options shares shall vest as follows: 2,008 shares on each of 3/26/2017 and 3/26/2018 and 2,010 shares on 3/26/2019.
Options shares granted from City Holding Company's 2013 Incentive Plan are subject to both time-based and performance-based vesting.  Options shares will vest as follows: 1,978 shares on 2/26/2019 and 1,980 shares on 2/26/2020.
Options shares granted from City Holding Company's 2013 Incentive Plan are subject to both time-based and performance-based vesting.  Options shares will vest as follows: 3,660 shares on each of 2/24/2019; 2/24/2020; and 2/24/2021.
Options shares granted from City Holding Company's 2013 Incentive Plan are subject to both time-based and performance-based vesting.  Options shares will vest as follows: 2,524 shares on each of 2/22/2020; 2/22/2021; and 2/22/2022.
Victoria A. Faw, attorney-in-fact
2020-02-28