What is the Purpose of Mastodon?
I'd say there are 4 main purposes of mastodon (and the rest of the fediverse).
The first purpose is the same purpose as tumblr or twitter etc. It's social media. It's meant for people to find other people, connect with them, and post and share content that matches their interests.
To decentralize social media. This is the big pull of mastodon that makes it very different from twitter, facebook and tumblr. It's also the feature that is most confusing to understand. I'll explain it more below. It also leads into #3 and #4.
To be a place where you can be in control of the rules. Don't want Nazis? You can ban them, or join a server that has strong rules against them.
To get away from commercialized forms of social media so you aren't constantly bombarded by ads, and your data isn't being fed to advertisers, and so you know the platform you are on won't cave to advertisers and do things like ban porn.
This is the easiest to understand: it's social media, we all know what that is. If people think the fediverse is only about #1, they go "ok but the internet is full of places to meet people, I don't need another website to do that on". And they'd be right! But it's the other three purposes that make the fediverse unique.
In my opinion this is the key to the fediverse. And it seems really complicated and too difficult to understand but it's really not.
Imagine you could download tumblr's code, then upload it to your own website, and run your own copy of tumblr. As a bonus, you could still follow anyone over on the main tumblr website and they could follow you. It would be exactly like using Main Tumblr; the experience would be seamless.
But there would be advantages too. Like if Main Tumblr decided to cave to advertisers and ban porn, for example. YOU wouldn't have to stop posting porn, because you're not on Main Tumblr, you're using your copy of Tumblr on your site, and your rules say porn is A-OK. So you could keep posting all the porn you wanted, and anyone who followed you could see it, and no one could stop you (as long as you're following the laws of the country your server is set in obviously).
And what if Main Tumblr decided to shut down? Not a problem for you - you're not using their site, so all your data is safe! If Discord goes down, every single Discord server also dies with it.
Twitter is dying; every single Twitter community and everyone's followings they've built are all dying with it because there's only one Twitter. A lot of artists and other people who depend on Twitter due to the followings and the communities/brands they've built on there are literally losing their livelihoods because Twitter is dying, and there's only one Twitter.
But if they'd been on a mastodon server that was dying, they could just move to a different server and all of their followers go with them - they wouldn't lose their audience and they wouldn't have to rebuilt it. Or, even better, they could just host their own server for their own brand and never have to worry about it going down on them, ever.
Now yes, hosting a website does cost money. And if you don't have money, that's ok! There are a bunch of servers out there already that are running and funded and accept users for free, and most of them have a set rule that if, for some reason, they have to close the server, you will get a 3 month warning to pick somewhere else to go.
All you have to do is read their rules and see if it's a community you jive with. Then after you join, you can follow anyone anywhere. It's that open, and that easy.
Another huge advantage of the fediverse is that every server can have its own rules, just like Discord servers can have their own rules. Many servers have extremely strict rules against nazis, racism, homophobia, spam, etc.
And the best part is, these server admins can actually enforce these rules because most of the servers are small. This means there's not that many people to deal with and moderation is actually a feasible task.
You might have heard "on facebook, you are the product" and it's true. Your data (and your identity) is the product being sold to the advertisers on those platforms. With no ads, there's no one trying to buy your data. The server your are on is the product, one you can choose to donate to if you wish.