The Easter Bunny Beginnings
The Easter Bunny has been around since the 1500's in the writings of the Germans. The first edible Easter bunny was made out of pastry and sugar in the early 1800's, and the bunny was said to lay colorful eggs in the nests that children made out of bonnets.
Germans who traveled to the Pennsylvania Dutch country brought the German Easter Bunny traditions with them to America in the 1700's.
Evolution of the Easter Bunny
Slowly the hats that the children piled into hidden nests for the bunnies turned into baskets. These baskets are still hidden around the house and are now used to collect the colored eggs of children as they go on their Easter egg hunts.
Other Easter Bunny Origins
Easter itself has a much different beginning than that of the Easter Bunny, but for that history you should check out "Is Easter a Christian or Pagan Holiday?"
The goddess Eostre in Pagan celebrations was associated with fertility, therefore also associating rabbits and eggs with the celebration. Over time, missionaries transformed these celebrations to be for religious reasons and the rabbits and eggs remained, even though their meanings have changed for most people.