The fediverse is a group of services (commonly social networking) that allow users to create accounts and talk (federate) outside of their instance to other instances in the network using an open communications standard.
The fediverse is better than standard, centralised social media as you can choose the server that stores your data or even start your own and talk to the rest of the network.
You can choose which software and which server you participate in, potentially allowing you to join a server which has a community with similar interests to you (e.g. a goth instance or a instance focused on metal and rock music)
There are three main standards of communication over the Fediverse which make up three networks.
This is the most common network/protocol which was defined by the World Wide Web Consortium in 2018.
It's used most commonly by servers running Mastodon, Pleroma and MissKey but most fediverse users will be using ActivityPub software.
This is a small open standard made by the team at diaspora*.
This is obviously used almost exclusively by diaspora*
This is the oldest means of federation which was implemented by Evan Prodromu in 2008.
This is most commonly used by Friendica and GNU social.
Many Fediverse users are on either ActivityPub instances or diaspora* pods.
If you really can't decide which one to use out of the two, either choose Mastodon (for ActivityPub) or find a Hubzilla instance which communicates with all three.