When you do something gaily, you do it in a happy, carefree way. If you're really glad that your cousins are visiting, you'll wave gaily to them as they pull up in your driveway.
This adverb is ideal for things done in a merry or blithe fashion, like when your dog trots gaily down the sidewalk, wagging his tail, or a bunch of kids laugh gaily together at a silly joke. It's all about happiness in the moment. The word comes from the Middle English gai, "joyful, happy, or pleasant." If vocabulary is your favorite thing, you'll gaily dive into word quizzes and games whenever you get the chance.