For those scared of all the links, here's a simple summary from someone who runs a server with an instance of one of these applications.
* ActivityPub is just an open, flexible protocol for sharing content
* The fediverse is a nice, human friendly way to describe all the various services using activitypub to talk back and forth to each other. The value in this is that these projects are diverse, it's not all twitter clones. You have video hosting services like peertube, the twitter-esque sites like mastodon and pleroma, open source managed file stores like nextcloud, and all of them can chat if they want to
* these services are hosted by whoever wants to host an instance, and they choose who they wish to federate with and who they don't, this optional interoperability is really the key feature for the entire network and protocol
Mastodon, as a twitter replacement is good from a user perspective (except in 300+ comment threads) , but I'd strongly recommend Pleroma at this point if you intend to run your own instance. The software of the two is similar, but Mastodon is in a constant state of chasing new, usually niche or controversial features which have led to a lot of bloat in terms of both management tooling, setup, and hardware required to run it.