No on knows exactly how this day got started, but it is widely believed that it had to do with the changing of the clander in 1582. Previous to 1582 many cultures, including those of the Romans and Hindus, celebrated New Year's Day on or around April 1. The reason they did this, was because April 1st closely follows the vernal equinox.
But, in 1582 Pope Gregory XIII ordered a new calendar (the Gregorian Calendar) to replace the old Julian Calendar. The new calendar called for New Year's Day to be celebrated January 1st. So that year many countries, including France, adopted the reformed calendar and moved New Year's day to January 1st. Well, apparently many people either refused to accept the new date, or did not learn about it since back then news traveled so slowly. As a result many people continued to celebrate New Year's Day on April 1. Other people that considered themselves to be more cultured, because they celebrated the "new" New Year's day, began to make fun of these traditionalists by sending them on "fool's errands" or trying to trick them into believing something that wasn't true.
Eventually, the practice spread throughout Europe and came to be known as April Fools' Day.