In 1563, the Council of Trent called for the western world to move from the Justinian calendar to the Gregorian calendar. (I'll let you look up why they called for this.) This meant moving the start of the new year from the Spring Equinox, around April 1st, to January 1st. In those days, news traveled slowly. When France finally adopted the change in 1582, some continued to celebrate the new year at the beginning of April. These people became known as April fools or "poisson d'avril" (April fish). And this is the origin of April Fool's day.
OR
In ancient Rome, followers of Cybele celebrated Hilaria, a festival at the end of March, by dressing in disguises and costumes, taunting and mocking important members of the community. This pagan celebration may have been adopted later by Europe at large. (Who doesn't like to roast the rich and powerful?)
OR
In Scotland, April Fool's Day may have been adapted from a two-day occurence beginning with Hunting of the Gowk (a foolish errand) and ending with Tailie Day (pinning tails or signs on someone's backside).
OR
April Fool's Day may have come from a celebration of the Vernal Equinox (the first day of Spring) when nature often fools us by not being very Spring-like.
Pick your theory. Or perhaps I should say, "Pick your poisson."