The Revenue Act of 1913 (PDF) which established Income Tax set the date as the 1st of March 1914 as the date to file taxes going forward - the PDF form is still available from the IRS if you haven't filed your 1913 taxes yet!

The date was moved in the Revenue Act of 1918 (PDF) to the 15th of March in order to give people more time to get taxes filed.

The current April 15th date was established by The Internal Revenue Code of 1954 - section 6072 (PDF) in order to (as evidenced by the House Ways & Means Chair Daniel Reed to the Joint Committee on Internal Revenue Taxation in 1953) give both taxpayers and the IRS more time.