Doing something jauntily means you're filled with cheer, style, and self-confidence while you do it. If you're excited about showing off an especially fashionable outfit, you might walk jauntily down the hallway at school.
If you've ever watched a dog show, you know the winner is inevitably a lively, handsome pup that trots jauntily around the ring. You could jauntily sing a little tune while you're taking out the trash or wear your hat tipped jauntily to one side of your head. This adverb implies a certain stylishness combined with good cheer. It comes from jaunty and its French root, gentil, "nice or pleasing."