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

    <schemaVersion>X0303</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2010-08-20</periodOfReport>

    <notSubjectToSection16>0</notSubjectToSection16>

    <issuer>
        <issuerCik>0000098537</issuerCik>
        <issuerName>TODD SHIPYARDS CORP</issuerName>
        <issuerTradingSymbol>TOD</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0000905516</rptOwnerCik>
            <rptOwnerName>HODGSON PATRICK W E</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>PO BOX 3806</rptOwnerStreet1>
            <rptOwnerStreet2></rptOwnerStreet2>
            <rptOwnerCity>SEATTLE</rptOwnerCity>
            <rptOwnerState>WA</rptOwnerState>
            <rptOwnerZipCode>98124</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>1</isDirector>
            <isOfficer>0</isOfficer>
            <isTenPercentOwner>0</isTenPercentOwner>
            <isOther>0</isOther>
            <officerTitle></officerTitle>
            <otherText></otherText>
        </reportingOwnerRelationship>
    </reportingOwner>

    <nonDerivativeTable>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>TODD SHIPYARDS CORP COMMON STOCK</value>
            </securityTitle>
            <transactionDate>
                <value>2010-08-20</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>A</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>750</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>15.18</value>
                    <footnoteId id="F1"/>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>A</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>77900</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
    </nonDerivativeTable>

    <derivativeTable>
        <derivativeHolding>
            <securityTitle>
                <value>TODD STOCK SETTLED APPRECIATION RIGHTS</value>
            </securityTitle>
            <conversionOrExercisePrice>
                <value>16.98</value>
                <footnoteId id="F2"/>
            </conversionOrExercisePrice>
            <exerciseDate>
                <footnoteId id="F3"/>
            </exerciseDate>
            <expirationDate>
                <footnoteId id="F4"/>
            </expirationDate>
            <underlyingSecurity>
                <underlyingSecurityTitle>
                    <value>COMMON STOCK</value>
                </underlyingSecurityTitle>
                <underlyingSecurityShares>
                    <value>2000</value>
                </underlyingSecurityShares>
            </underlyingSecurity>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>2000</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </derivativeHolding>
        <derivativeHolding>
            <securityTitle>
                <value>TODD STOCK SETTLED APPRECIATION RIGHTS</value>
            </securityTitle>
            <conversionOrExercisePrice>
                <value>15.18</value>
                <footnoteId id="F5"/>
            </conversionOrExercisePrice>
            <exerciseDate>
                <footnoteId id="F6"/>
            </exerciseDate>
            <expirationDate>
                <footnoteId id="F7"/>
            </expirationDate>
            <underlyingSecurity>
                <underlyingSecurityTitle>
                    <value>COMMON STOCK</value>
                </underlyingSecurityTitle>
                <underlyingSecurityShares>
                    <value>2000</value>
                </underlyingSecurityShares>
            </underlyingSecurity>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>2000</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </derivativeHolding>
    </derivativeTable>

    <footnotes>
        <footnote id="F1">Fair market value on date of election. Shares will vest ratably over the next 36 months and are subject to forfeiture to the Company prior to vesting under certain circumstances. Granted as restricted shares under Todd's 2003 Incentive Stock Plan.</footnote>
        <footnote id="F2">Upon exercise of the SSARs, the holder is entitled to receive value equal to the number of rights being exercised times the then current fair market value reduced by $16.98, the fair market value on date of grant. Such value will be paid, net of withholding taxes, in shares of Todd Common Stock.</footnote>
        <footnote id="F3">One-third of the 2,000 SSARs vest each year on August 21 commencing in 2010 (through 2012).</footnote>
        <footnote id="F4">Upon certain termination events, but no later than August 21, 2014.</footnote>
        <footnote id="F5">Upon exercise of the SSARs, the holder is entitled to receive value equal to the number of rights being exercised times the then current fair market value reduced by $15.18, the fair market value on date of grant. Such value will be paid, net of withholding taxes, in shares of Todd Common Stock.</footnote>
        <footnote id="F6">One-third of the 2,000 SSARs vest each year on August 20 commencing in 2011 (through 2013).</footnote>
        <footnote id="F7">Upon certain termination events, but no later than August 20, 2015.</footnote>
    </footnotes>

    <remarks></remarks>

    <ownerSignature>
        <signatureName>PATRICK W.E. HODGSON</signatureName>
        <signatureDate>2010-08-20</signatureDate>
    </ownerSignature>
</ownershipDocument>
