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

    <schemaVersion>X0306</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2013-08-08</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>
        <nonDerivativeHolding>
            <securityTitle>
                <value>Common Shares, No Par Value</value>
            </securityTitle>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>83329</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>I</value>
                </directOrIndirectOwnership>
                <natureOfOwnership>
                    <value>See footnotes</value>
                    <footnoteId id="F1"/>
                    <footnoteId id="F2"/>
                </natureOfOwnership>
            </ownershipNature>
        </nonDerivativeHolding>
        <nonDerivativeHolding>
            <securityTitle>
                <value>Common Shares, No Par Value</value>
            </securityTitle>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>25295</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeHolding>
    </nonDerivativeTable>

    <derivativeTable>
        <derivativeTransaction>
            <securityTitle>
                <value>Option to Purchase Common Shares</value>
            </securityTitle>
            <conversionOrExercisePrice>
                <value>43.92</value>
            </conversionOrExercisePrice>
            <transactionDate>
                <value>2013-08-08</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>A</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>4657</value>
                    <footnoteId id="F3"/>
                </transactionShares>
                <transactionPricePerShare>
                    <value>0</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>A</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <exerciseDate>
                <footnoteId id="F4"/>
            </exerciseDate>
            <expirationDate>
                <value>2023-08-08</value>
            </expirationDate>
            <underlyingSecurity>
                <underlyingSecurityTitle>
                    <value>Common Shares, No Par Value</value>
                </underlyingSecurityTitle>
                <underlyingSecurityShares>
                    <value>4657</value>
                </underlyingSecurityShares>
            </underlyingSecurity>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>20482</value>
                    <footnoteId id="F5"/>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </derivativeTransaction>
        <derivativeTransaction>
            <securityTitle>
                <value>Career Restricted Stock Units</value>
            </securityTitle>
            <conversionOrExercisePrice>
                <footnoteId id="F6"/>
            </conversionOrExercisePrice>
            <transactionDate>
                <value>2013-08-08</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>A</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>3016</value>
                    <footnoteId id="F7"/>
                </transactionShares>
                <transactionPricePerShare>
                    <value>0</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>A</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <exerciseDate>
                <footnoteId id="F8"/>
            </exerciseDate>
            <expirationDate>
                <footnoteId id="F8"/>
            </expirationDate>
            <underlyingSecurity>
                <underlyingSecurityTitle>
                    <value>Common Shares, No Par Value</value>
                </underlyingSecurityTitle>
                <underlyingSecurityShares>
                    <value>3016</value>
                </underlyingSecurityShares>
            </underlyingSecurity>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>3016</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </derivativeTransaction>
    </derivativeTable>

    <footnotes>
        <footnote id="F1">Richard C. Breeden is the managing member of Breeden Capital Partners LLC and the 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. (the &quot;Fund&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 Fund; however, he disclaims beneficial ownership of such shares except to the extent of his pecuniary interest therein.</footnote>
        <footnote id="F3">On August 8, 2013, the Issuer granted 4,657 stock options to Mr. Breeden as compensation for services rendered as a director of the Issuer.  Pursuant to the governing documents of the Fund, any economic interests realized with respect to these stock options will be distributed to the Fund.</footnote>
        <footnote id="F4">The stock options reported herein are fully vested immediately.</footnote>
        <footnote id="F5">Includes previously granted stock options with different terms of exercisability.  Pursuant to the governing documents of the Fund, any economic interests realized with respect to these stock options will be distributed to the Fund.</footnote>
        <footnote id="F6">Each Career Restricted Stock Unit represents the right to receive one share of Common Stock six months after the cessation of Mr. Breeden's service as a director of the Issuer.</footnote>
        <footnote id="F7">On August 8, 2013, the Issuer granted 3,016 Career Restricted Stock Units of the Issuer to Mr. Breeden as compensation for services rendered as a director of the Issuer.  Pursuant to the governing documents of the Fund, any economic interests realized with respect to these Career Restricted Stock Units will be distributed to the Fund.</footnote>
        <footnote id="F8">Each Career Restricted Stock Unit is fully vested immediately and will be settled in shares of Common Stock six months after the cessation of Mr. Breeden's service as a director of the Issuer.</footnote>
    </footnotes>

    <remarks></remarks>

    <ownerSignature>
        <signatureName>/s/ Richard C. Breeden</signatureName>
        <signatureDate>2013-08-12</signatureDate>
    </ownerSignature>
</ownershipDocument>
