The short version is as follows:
Mastodon is a Twitter-like social network, and the Fediverse is a network of social networks.
But let's also delve into the slightly longer version below.
If you don't like how Mastodon is being run, you can go to another Mastodon instance. That's right - there are multiple Mastodon instances, all being run slightly differently by different people. If you're being harassed on Twitter and Twitter won't help you, you're out of luck - there's no "other Twitter" to go to. But Mastodon is different. There are many Mastodon instances, all run by different people. But the Mastodon instances are connected in a giant network called the Fediverse (more on that below), which means that if you have an account on one Mastodon instance, you can still follow and interact with people who have accounts on another instance. Each Mastodon instance has its own URL - the most popular and well-known one is Mastodon.social, but there are many others. Oh, and by the way, if you're confused by the ".social" part of the URL, that's just another top level domain now, just like .com and .org.
You should use Mastodon because it's truly free. Mastodon isn't run by a corporation to make money. It is run by hobbyists (typically) because they enjoy it. They're not trying to make money off your data, they don't track you, and there are no ads.
Each instance typically has a theme, or a topic. If you're into technology in general, there's for example one called Mastodon.technology. There are Mastodon instances for various social groups, and languages. If you go to joinmastodon.org you can find a list.
Personally, I'm partial to the Instance I'm running targeted at Python developers of all stripes - PythonDevs.social (although it's not strictly a Mastodon instance, it runs another software called Pleroma which is compatible with Mastodon).
Yes, you can use Mastodon (and Pleroma, as in the case of PythonDevs.social, since it's compatible with Mastodon) various apps. When it comes to Android, I would recommend Tusky, but you can search the App Store and Play Store for "Mastodon", and you'll find a variety of available apps. Also, you can use the web interface on your smartphone as well as your desktop by simply clicking on the link to an instance, such as PythonDevs.social.
Okay. By now, I hope we all at least sort of understand what Mastodon is. So what's this about Fediverse? What's that?
I'm glad you asked. Let's say you have a Twitter account, and you want to follow your friend. But she doesn't have a Twitter account, she only has a Facebook account - and for one reason or another, you don't want a Facebook account, and she doesn't want a Twitter account.
That means you simply can't follow eachother. Such is life. The end.
The Fediverse is different. As I said initially, the Fediverse is a network of social networks. For example, Mastodon instances are connected to the Fediverse. PeerTube, an open source alternative to YouTube, is also connected to the Fediverse, as is Friendica which is more like Facebook than Twitter. This means that any user on any of these instances can follow and interact with any other user on any other instance, as if a Twitter user could follow a Facebook user.
First of all, you should find an instance where you agree with the Terms of Service (they're typically short, and easy to understand). For example, you can find the Terms of Service on PythonDevs.social by clicking About on its start page.
You can report users who misbehave, just like on any other social network. Most instances are small, and have active administrators who care about dealing with such things. But what if there's an instance which doesn't care if their users abuse the rest of us? Then what do we do?
Introducing Gab. Gab is Fediverse instance which values free speech over everything else. They let their users pretty much say and do anything they want, including harassing other people. This means that nazis, homophobes, bullies, and other lowlives flock to Gab (I think of Gab as a bridge, under which the trolls live). At the risk of sounding like Trump, I'll add that I assume that there are some good people on Gab as well who simply value free speech over the ability to be an unfettered asshole, but I'd wager they're in the minority.
This unwillingness of dealing with trolls on the part of the Gab administration posed a problem for the Fediverse, which most of us instance administrators simply dealt with by blocking all content from Gab on our instances. This means that while Gab is technically part of the Fediverse, almost all Fediverse instances refuse to communicate with them. We simply leave them alone to stew in their echo chamber of hate, homophobia, and general assholery. At least we don't have to see any of it - unless we choose to join an instance which doesn't block Gab, or even Gab itself.
The Fediverse is a ghost town! I don't know anyone there! All my friends are on Twitter and Facebook!
Sure. But instead of finding your friends in the Fediverse, you can make friends with the people that are there! Again, find an instance that caters to your interest, and start talking to people on that instance. And keep at it - keep coming back, and you'll find that you have meaningful interactions with like-minded people every time you do.
If this post has piqued your interest, and if you're into Python development (which I assume you are since you're on pythonvarmland.se), then I encourage you to come join us at PythonDevs.social. The instance is focused on Python development, but it's not just for Python developers from Värmland - it's open to everybody. You'll find me at PythonDevs.social/Enfors. In a future blog post, I'll go into a little more detail about this particular instance, and what it can offer Python developers.