We see the very first Christmas presents being given far back in history. In ancient Rome, during the feast of Kalends, it was tradition for the high-ranking officials to give gifts to the Roman Emperor. This wasn't technically a Christian holiday (it was actually pagan) but it does coincide with the time of year and the gift-giving tradition.
A person who jump-started the gift-giving traditions at Christmas was good old Saint Nicholas. He was known for giving gifts.
Christmas presents as we know them today actually started in America. It's been a longtime tradition to give little gifts to friends and family, but the massive giving that goes on nowadays is a creation all our own.
In the early 1800s stores and companies started advertising Christmas gifts. By the 1840s, the tradition of lavish giving was fully ingrained into American society.