The fediverse is a federated universe of servers commonly used for sharing content, like social media. So, instead of having to rely on a single organization to run the server (e.g. Facebook), the fediverse is a giant collection of servers across the world, owned by many people and organizations.
Take a look at this depiction of a federated network - each server in this photo is owned and run by different administrators/owners. Federated networks are best explained as email servers - you have an email account that exists on a server (e.g. Outlook), your friend has an account on a different server (e.g. GMail), and another friend has an account on a third server (e.g. ProtonMail). All three of you can talk and communicate back and forth without having to be on the same server. However, responsible email admins are there to set rules and control the traffic going in/out of the server.
The main objective of this architecture is to decentralize the control within the internet connections. For example, if you run your own Mastodon instance, you and your users can't be censored or impacted in any way by authorities of another Mastodon instance. Some users have praised these features due to recent criticism of popular social media websites that may be over-censoring their users.
This strategy is great for making sure control of the social web isn't controlled by a single organization, but it also has some downsides. If I create a Mastodon instance and get a ton of users to sign up, I can shut the server down at any time. That means you're at risk of losing the content you've created unless you back it up, or the server backs it up for you. Also, depending on the software used (e.g. Mastodon, Pixelfed, etc.), censorship may still be an issue if the server admins decide they want to censor their users. Now, censorship isn't always a bad thing and can even benefit the community as a whole, but you'll want to determine which servers align with your idea of proper censorship.
However, these are risks that we take when we sign up for any online platform. Whatever your reason is for trying out federated social networks, they are part of the future of the internet. However, the popularity of these services is still yet to be determined, especially with the increased difficulty understanding and signing up for these platforms. Perhaps increased regulation and litigation against current social media sites will push more users into the fediverse.
The list below is a small guide that will show you federated alternatives to current popular websites. There are many more out there, so feel free to send me an email if you have suggestions or want your website included.
The best way to get started is to simply sign up and learn as you go. If you're comfortable signing up through a Mastodon, Pleroma, or Friendica server, here is a list of themed servers to choose from. If you're looking for something else, try a web search for a federated alternative to your favorite sites.
Find a server that focuses on your passions and start there!