Well in our family that's two different games.
Dirty santa meaning that all gifts are good gifts. They are given a number and put under the tree. Everyone draws a number. #1 gets the gift labeled #1. #2 gets the gift labled #2 and can keep it or trade with #1, so on and so forth until back to #1 who gets the last trade if they choose.
White elephant meaning you play dirty santa, but by taking a gift you have received in the past but didn't want and you "regift" it.
We love doing that. This year however, we have decided to draw names at Christmas only. Then the rest of the year, you send that person presents or cards or whatever you feel like doing for them all year. Kind of a secret santa type thing except with a family member. That way, we're not doling out money at Christmas, when it can be kind of tight, and we can spend more during the whole next year if we choose.