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

    <schemaVersion>X0305</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2012-04-23</periodOfReport>

    <notSubjectToSection16>0</notSubjectToSection16>

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

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001186066</rptOwnerCik>
            <rptOwnerName>DAVIS R RUSSELL</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>0</isDirector>
            <isOfficer>1</isOfficer>
            <isTenPercentOwner>0</isTenPercentOwner>
            <isOther>0</isOther>
            <officerTitle>Sr Vice President</officerTitle>
            <otherText></otherText>
        </reportingOwnerRelationship>
    </reportingOwner>

    <nonDerivativeTable>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Stock, $1 par</value>
            </securityTitle>
            <transactionDate>
                <value>2012-04-23</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>F</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
                <footnoteId id="F1"/>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>1556</value>
                </transactionShares>
                <transactionPricePerShare>
                    <footnoteId id="F2"/>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>48656.9109</value>
                    <footnoteId id="F3"/>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
    </nonDerivativeTable>

    <derivativeTable></derivativeTable>

    <footnotes>
        <footnote id="F1">Payment of tax liability by withholding securities incident to the receipt of shares granted under the Company's 2000 Long-Term Incentive Compensation Plan for which restrictions lapsed 04/23/2012.</footnote>
        <footnote id="F2">The number of shares withheld was based on the closing price of the Company's common stock on 04/23/2012 ($39.58).</footnote>
        <footnote id="F3">Total includes 29.8812 shares acquired through contributions and/or reinvestment of dividends under the Company's 401(k) plan during the period 01/01/2012 through 04/23/2012.</footnote>
    </footnotes>

    <remarks></remarks>

    <ownerSignature>
        <signatureName>Julia E. Callis, Attorney-in-Fact for Russell Davis</signatureName>
        <signatureDate>2012-04-24</signatureDate>
    </ownerSignature>
</ownershipDocument>
