April Fools Day has some mysterious origins but there is one pretty interesting theory for pranks and jokes on 1st April.
In 13th century England, King John decided to acquire some of the land of Gotham in Nottinghamshire for a hunting lodge. This idea was not popular with the townsfolk (then again, King John wasn't in any way popular). So to dissuade the king, the townsfolk decided to come up with a cunning plan. The plan was to play the fool where the townspeople would be found by the king's men doing all sorts of crazy things like trying to drown fish. When the kings men arrived in the town and saw the antics, they decided to counsel the king to choose somewhere else for his hunting lodge as the town was filled with mad (as in crazy) people.
It wasn't until 1686 when John Aubrey referred to Fooles Holy Day and by that time, April Fool traditions had already been established across Britain.
As for shouting April Fool when someone has been tricked on 1st April, that may have been established in France where in 1582, France switched from the Julian calendar to the Gregorian calendar. In the Julian calendar as well as the Hindu calendar, New Year's Day began on 1st April instead of 1st January like in the Gregorian calendar. As a result, people were slow at adjusting to the new calendar still celebrated New Year's during the last week of March through 1st April and were called April Fools as a result. Such pranks included having paper fish placed on their backs and being called "poisson d'avril" (April Fish).