It is thought that it was born out of the confusion that resulted from the introduction of the Gregorian calendar. Back in 1582, France's King Charles IX made the change from the Julian calendar of ancient Rome to the Gregorian calendar. In so doing, New Year's day, which had been celebrated between March 25 and April 1 was moved to January 1. As communication was pretty slow back then, there were plenty of people who did not get the memo and proceeded with their New Year's celebrations on April 1st. They were deemed fools and the tradition of coming up with elaborate pranks to trick others into seeming foolish was born.
While commonly believed to be the original origin of the day this theory does not account for the fact that traditions of exuberance and tomfoolery during this time of year have existed for thousands of years. It seems that the turning of the seasons and beginning of spring is always a time for some sort of celebration and the changing of the calendars was a great way to sort of codify a day for general merriment and mischief.
The April Fool's tradition takes different forms in different countries. In France, Belgium, and the Netherlands, it's tradition to try to stick a paper fish onto a friend's back without him noticing and then calling out "April fish!" In Scotland you try to fool friends into a mission seeking out a fool, all while making your friend look foolish. "Kick me" signs stuck to a person's back can be traced back to Scottish April Fools' celebrations!
I always enjoy seeing what fun ideas companies like Google come up with. Here's a list of their pranks over the years. Here's a list of their jokes for 2013. And here's a collection of the best April Fools' jokes over the years. The spaghetti tree is the best!