Specifically, Mastodon is one server that speaks the ActivityPub standard. In the same way that Matrix is a protocol that specifies clients (like Riot) and servers (like Synapse), ActivityPub is a protocol that is designed to support many kinds of social networking in a distributed fashion. Mastodon is the leading service built on ActivityPub. As with Synapse, you can run your own Mastodon server and communicate with people on other Mastodon servers via federation. It's been getting more notice lately following allegations of censorship on Twitter from the political right. Whether or not these are correct, Twitter still has the power to accomplish censorship; in a distributed network, no one entity has the ability to censor particular users, posts, or viewpoints.
The big social networks of today also have bad incentive structures. They make money via targeted advertising across the web. They make more money the more time you spend on their applications. And because of the way we react to ideas, this produces the clearly bad incentive to promote outrage, resulting in a more hateful and divided population.
These incentives don't exist with a distributed, federated network like Mastodon (which appears to be its primary name, though ActivityPub might be the more technically correct name for the network). Users run nodes and pay for them, setting the network up to serve the users rather than the other way around.
Right now, there's a relatively high barrier to entry mostly due to network effects. I don't know who's on Mastodon who I'd follow (though I'm intending to set up my own instance soon to try it out; I suppose I'll find out). The technical barriers are not high - you can join any node and follow anyone, making the typical complaint (the length of the list on JoinMastodon.org) not as strong as it could be. You don't have to start your own node in order to be on Mastodon. Supposedly there are issues with finding interesting people to follow - the search feature is local to each node (which sounds very limiting, but it means it will include any users anyone on that node follows, so it's not bad).
I'll keep you updated as I try it out.