Easter Bunny is a fantasy character which seems to be originated in Germany. In German, Easter bunny is called Osterhase and the tradition dates back to 1700's. According to the tradition, on the day of Easter the Osterhase lays eggs and decorates them and hides the eggs in gardens for the kids to find out. It is believed that it is the German immigrants who came to America in the 1700's brought the concept of the Easter bunny in America. There are mentions about the Easter bunny character as early as 1500 and it is thought that they became popular in the 1700's. In one of the notes about Easter bunnies which dates back to 1684, a professor in medicine mentioned about the side effects of overeating Easter eggs.
Rabbits and Hares were always associated with fertility and spring, and hence they became a symbol of birth and life as well. This is the reason why bunnies got linked with Easter where people celebrate the resurrection of Jesus Christ. The fictional character of Easter Bunny lays eggs and fill the Easter baskets for children, this is what the children are told. In reality, we know rabbits don't lay eggs. However, this belief is also linked with few legends from history which are related to the Anglo - Saxon goddess of spring whose name was Eostre ( or Ostara).
The below image from Wikimedia Commons Image is that of an Easter Bunny postcard from 1907.