Your Periscope username is your identity on the app. It’s what people will use to search for you as well as what other users see when you chat on broadcasts.
Upon sign up, a list of usernames will be suggested based on your Twitter handle if you signed up using Twitter, or you can create a different username of your choice. Your username must be unique and have at least two characters. If the username icon turns red upon creation, the username is unavailable.
Related: How do I change my username?