Secret Santa is when everyone pulls the name of another participant and buys that specific person a gift. Usually you try to get them something they genuinely want.

White Elephant/Yankee Swap is when everyone brings a gift without a specific recipient in mind, everyone pulls a number, and you pick a gift in that order and can choose to either keep it or force a previous opener to swap with you. A higher number is better because you have more to choose from, except for #1 who gets their pick of anything at the end. The gifts often tend to be on the silly side.