Badges are a mark of achievement when completing an Event, Solo, or Weekly challenge. Many of the challenges for the badges can be something like scoring a certain amount of points in a game, completing a solo challenge, or completing events. To view all your badges and see which new ones can be earned, go to your profile and tap on Recent Badges. There it will show a list of badges and the date each was earned.
At the top, you can tap on the slideshow displaying current badges that can show how to obtain it, how much time there is left to do so, and maybe some rewards that come with it. Words With Friends Badges are a great way to view your progress in the game and see which type of challenges you finish. Plus, it can be a fun way to show off your accomplishments to your friends.