Hey, great question!
Nostr is another open communication protocol, like ActivityPub. It's short for: "Notes and Other Stuff Transmitted Over Relays". The key differences with Nostr are two things:
Instead of an inbox/outbox model, Nostr relies on relays and clients. Basically, you register a keypair representing your online identity, and you use a secret key (known as NSec) to log in through a client. The client then loads interactions directed at you through a relay, and your interactions then get posted out from the client back to the relay.
Pretty much all interactions with the Nostr network are done through rich client applications. You don't really log into an instance, in the traditional sense, you just have a full-featured app do all of the interaction part for you. It's common to have a private wallet as a browser extension for handling app authentication, as opposed to OAuth built in to a server instance.
It's kind of hard to measure the overall userbase of the Nostr network. I would estimate that it's much smaller than Bluesky or the Fediverse, but it has an active developer ecosystem and user community.
