Of course, it was the Three Wise Men bringing gifts to the baby Jesus upon his birth which first sparked the occasion of gift giving.
The first known acts of giving presents around the time of Christmas was in Roman times, when they used to share gifts at the festival of Kalends.
Hanging stockings is said to have started from St Nicholas, who wanted to help out a family of three girls who had no money to get married. He crept down their chimney and placed a bag on gold in each of the girls' stockings, which were hung by the fireplace to dry.