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

    <schemaVersion>X0306</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2013-03-06</periodOfReport>

    <notSubjectToSection16>0</notSubjectToSection16>

    <issuer>
        <issuerCik>0000815065</issuerCik>
        <issuerName>STERIS CORP</issuerName>
        <issuerTradingSymbol>STE</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001300622</rptOwnerCik>
            <rptOwnerName>Breeden Richard C</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>C/O BREEDEN CAPITAL MANAGEMENT LLC</rptOwnerStreet1>
            <rptOwnerStreet2>100 NORTHFIELD STREET</rptOwnerStreet2>
            <rptOwnerCity>GREENWICH</rptOwnerCity>
            <rptOwnerState>CT</rptOwnerState>
            <rptOwnerZipCode>06830</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>Common Shares, No Par Value</value>
            </securityTitle>
            <transactionDate>
                <value>2013-03-06</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>J</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
                <footnoteId id="F1"/>
                <footnoteId id="F2"/>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>565533</value>
                </transactionShares>
                <transactionPricePerShare>
                    <footnoteId id="F3"/>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>1129180</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>I</value>
                </directOrIndirectOwnership>
                <natureOfOwnership>
                    <value>See footnotes</value>
                    <footnoteId id="F1"/>
                    <footnoteId id="F2"/>
                    <footnoteId id="F3"/>
                </natureOfOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
        <nonDerivativeHolding>
            <securityTitle>
                <value>Common Shares, No Par Value</value>
            </securityTitle>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>25295</value>
                    <footnoteId id="F4"/>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeHolding>
    </nonDerivativeTable>

    <derivativeTable></derivativeTable>

    <footnotes>
        <footnote id="F1">Richard C. Breeden is the managing member of Breeden Capital Partners LLC, managing member and chairman and chief executive officer of Breeden Capital Management LLC.  Breeden Capital Partners LLC is in turn the general partner of Breeden Partners L.P., Breeden Partners (California) L.P. (the &quot;California Fund&quot;)  collectively with Breeden Partners L.P., the &quot;Funds&quot;).</footnote>
        <footnote id="F2">Pursuant to Rule 16a-1(a)(2)(ii)(B) of the Securities Exchange Act of 1934, as amended, Mr. Breeden in his capacity as managing member, as well as chairman and chief executive officer of Breeden Capital Management LLC, and as the managing member of Breeden Capital Partners LLC, may be deemed to be the indirect beneficial owner of the shares of common stock of the Issuer (the &quot;Common Stock&quot;) owned by the Funds; however, he disclaims beneficial ownership of such shares except to the extent of his pecuniary interest therein.</footnote>
        <footnote id="F3">On March 6, 2013, the California Fund distributed 574,863 shares of Common Stock to its investors for no consideration, including 9,330 shares to Breeden Capital Partners LLC.  Mr. Breeden did not directly receive any shares of Common Stock in the distribution.</footnote>
        <footnote id="F4">Mr. Breeden continues to hold of record 25,295 shares of Common Stock directly.</footnote>
    </footnotes>

    <remarks></remarks>

    <ownerSignature>
        <signatureName>/s/ Richard C. Breeden</signatureName>
        <signatureDate>2013-03-07</signatureDate>
    </ownerSignature>
</ownershipDocument>
