They are cosmetic indicators on your Nostr profile. You can get them by playing Thunder Games and linking your Nostr profile, joining certain relays, attending certain events, or you can create your own. https://badges.page is where people go to create, assign, and accept/reject them.

Not all clients show them. Only 3 or 4 do? I hope more clients show them though.

I have way too many badges 😆 but it's a lot of fun to collect them.