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

    <schemaVersion>X0306</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2013-08-15</periodOfReport>

    <notSubjectToSection16>0</notSubjectToSection16>

    <issuer>
        <issuerCik>0001089819</issuerCik>
        <issuerName>CLECO CORP</issuerName>
        <issuerTradingSymbol>CNL</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001186052</rptOwnerCik>
            <rptOwnerName>GARRETT J PATRICK</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>P.O. BOX 5000</rptOwnerStreet1>
            <rptOwnerStreet2></rptOwnerStreet2>
            <rptOwnerCity>PINEVILLE</rptOwnerCity>
            <rptOwnerState>LA</rptOwnerState>
            <rptOwnerZipCode>71361-5000</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>1</isDirector>
            <isOfficer>0</isOfficer>
            <isTenPercentOwner>0</isTenPercentOwner>
            <isOther>0</isOther>
            <officerTitle></officerTitle>
            <otherText></otherText>
        </reportingOwnerRelationship>
    </reportingOwner>

    <nonDerivativeTable>
        <nonDerivativeHolding>
            <securityTitle>
                <value>Common Stock, $1.00 par</value>
            </securityTitle>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>50732.808</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeHolding>
    </nonDerivativeTable>

    <derivativeTable>
        <derivativeTransaction>
            <securityTitle>
                <value>Phantom Stock</value>
            </securityTitle>
            <conversionOrExercisePrice>
                <footnoteId id="F1"/>
            </conversionOrExercisePrice>
            <transactionDate>
                <value>2013-08-15</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>A</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>474</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>46.74</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>A</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <exerciseDate>
                <footnoteId id="F1"/>
            </exerciseDate>
            <expirationDate>
                <footnoteId id="F1"/>
            </expirationDate>
            <underlyingSecurity>
                <underlyingSecurityTitle>
                    <value>Common Stock, $1.00 par</value>
                </underlyingSecurityTitle>
                <underlyingSecurityShares>
                    <value>474</value>
                    <footnoteId id="F1"/>
                </underlyingSecurityShares>
            </underlyingSecurity>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>61401.866</value>
                    <footnoteId id="F2"/>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </derivativeTransaction>
    </derivativeTable>

    <footnotes>
        <footnote id="F1">The shares of phantom stock relate to dividend reinvestment on shares of phantom stock pursuant to the terms and provisions of the Cleco Corporation Deferred Compensation Plan (the &quot;Plan&quot;).  The reporting person has elected to defer receipt of these shares under the Plan.  Each share of phantom stock is the economic equivalent of one share of Cleco Corporation common stock.</footnote>
        <footnote id="F2">Includes 13,119 shares of phantom stock (not subject to restrictions) related to dividend reinvestment on shares of phantom stock pursuant to the terms and provisions of the Plan.  Includes (a) 2,140 shares of phantom stock subject to restrictions that lapse on January 25, 2014; (b) 2,544 shares of phantom stock subject to restrictions that lapse on January 30, 2015; (c) 2,149 shares of phantom stock subject to restrictions that lapse on January 29, 2016; (d) 1,898 shares of phantom stock subject to restrictions that lapse on January 28, 2017; (e) 2,113 shares of phantom stock subject to restrictions that lapse on January 27, 2018 and (f) 1,964 shares of phantom stock subject to restrictions that lapse on January 24, 2019.</footnote>
    </footnotes>

    <remarks></remarks>

    <ownerSignature>
        <signatureName>Julia E. Callis, Atty-in-fact for J. Patrick Garrett</signatureName>
        <signatureDate>2013-08-16</signatureDate>
    </ownerSignature>
</ownershipDocument>
