The Fediverse is a network of interconnected servers (instances) that use decentralized protocols for communication. It encompasses various social media platforms and services, each running on its own server but able to interact with users on other servers. The key concept here is federation, where users from different instances can communicate with each other seamlessly.
It is basically like emails, there are plenty of providers out there (yahoo.com, gmail.com, hotmail.com etc ...) but they all manage to send each other messages.
Mastodon is one of the most popular platforms in the Fediverse. It's a decentralized microblogging platform (similar to Twitter/X). Users in the Fediverse can post short messages (toots), follow others, and engage in conversations. Mastodon instances are independent servers, each with its own rules and community, but users can interact with people on other instances.
Nostr is another (more recent) decentralized social platform, (similar to Mastodon) but with some differences in architecture and design. It uses a new protocol for transmitting data between servers (relays), making it lightweight and efficient. Nostr focuses on decentralization, privacy, and censorship resistance, allowing users to communicate without relying on a central authority.
Compared to mastodon, you do not need to signup when connecting to another server (instance/relay) ; the same nsec key is shared on all relays, making a smooth transition between all services. And second big change is the micro-payment (zap) featured in Nostr, which can serve as incentive to be active on the platform.