From the Medium blog post:
Rolling out to Medium users over the coming week will be a new, more satisfying way for readers to give feedback to writers. We call it “Claps.” It’s no longer simply whether you like, or don’t like, something. Now you can give variable levels of applause to a story. Maybe clap once, or maybe 10 or 20 times. You’re in control and can clap to your heart’s desire.
Medium plans to use the number of claps to determine how much to pay writers on their platform.