If you do something willingly, you're perfectly happy to do it. Invite your friends to join you for hot fudge sundaes, your treat, and they'll willingly come along.
When a person acts willingly, they are doing something of their own free will — in other words, they're glad to do it. Sometimes it's a no-brainer: you'll willingly jump in your aunt's waiting car when she offers to drive you home from school on a rainy day. At other times, doing something willingly is a marker of your good nature, like when you willingly step in to help your friend move.