In 1913 Congress passed the Sixteenth Amendment to the Constitution, which said that Congress can collect income tax. They’d already done this before, but were making it all official (although apparently when it was introduced it was expected to fail). At that point, they declared March 1 to be Tax Day. March 1, as you may recall, was New Year’s Day for the Ancient Romans, but that’s probably just a coincidence since quite a few days were New Year’s Day at one time and place or another.
After 1918 the Tax Day was changed to March 15, which probably resulted in a host of “Beware the Ides of March” jokes.
Finally, in 1954, the date was changed to April 15 [although that didn’t take effect until the following year]. According to this site (where I got most of the info so far), this was because the IRS was getting swamped with so many returns at the last minute, and they hoped that having more time would spread that out a bit.
In searching around, I also discovered the original 1040 form from 1913, which doesn’t look as simple as I would have hoped. You can see all of them through the years here, although just looking at all those tax forms doesn’t exactly give one a feeling of peace and relaxation.