This is written assuming you are moving from Discord to Matrix. If you've never used Discord in your life, I envy you.
IN SHORT: Matrix is cool and you can have free stickers and emojis. Sign up for an account on Cinny. For homeserver, you can stick with matrix.org. If you're an Eggware Social subscriber you can use chat.eggware.social but you'll have to contact me for the special registration code.
What?
At its simplest, Matrix is a free and open source alternative to Discord.
It's decentralized and federated, which means you're not relying on a single company.
The idea of "using Matrix" is made up of three components: the protocol, the server, and the client. The most appropriate comparison I can make is to IRC, but here's one you probably use every day: e-mail is a protocol, @gmail.com is a server, and mail.google.com is a client.
Matrix is the protocol. There are a few different choices for server software, but you don't have to worry about that if you're not self-hosting. There are a fair amount of clients, and they all have their own special problems.
Spaces and rooms
Matrix has spaces and rooms. This is roughly analogous to Discord's servers and channels, but there are some differences.
First off, every "channel" in Matrix is a room. Direct messages are 2-person rooms. Spaces are rooms... that contain other rooms.
Two spaces can contain the same room. There's a good explanation for that here. If you had a space for Garfield and a space for food, they could both share the Garfield Food room, and then you'd get some crossover from people who might otherwise not talk. This also cuts down on the need for sending people temporary invites to your friends-only server.
Joining a space doesn't automatically join all the rooms within it. This seems annoying, but I think it's a great idea. My most consistent problem with moderating Discord servers was that there wasn't an easy way to keep people out of channels that they weren't interested in.
Why?
Oh boy! This is the part where I get to sound like a walking advertisement!
Matrix offers end-to-end encryption. That means your data can stay private.
It's not proprietary, and development isn't driven by money. That means the community is generally the ones deciding what features get in.
Since Matrix officially supports community clients, you can install whatever accessibility features you need. Using third-party Discord clients is technically against the TOS even if you're just trying to make the service less hostile towards you.
The protocol doesn't require government-issued IDs for age verification. That means you can join channels with sensitive topics and you don't have to worry about your ID getting leaked in a data breach.
You can have all of the custom emojis that you want, and you can use them across servers for free. You can also use these emojis at sticker size!
How?
You will first need to choose a server and a client. I'll detail those soon.
The registration page on your client of choice will ask for a homeserver, a username, and a password. Depending on your server of choice, it might also ask for a registration token, essentially a secret key; hopefully your admin will have already provided you this.
Once you're signed in, you're free to explore the app a bit. You could make a new Room and try out emojis. Or you could join an existing room and embarrass yourself. Maybe you'd like to join the Eggware Social space!
Servers
If you're just looking to try things out, the official matrix.org homeserver is a decent enough starting point. I used it while I was starting out, and it worked fine. It has file transfer limitations you might want to know about, which is to be expected from the largest and official-est server.
If you're already an Eggware Social subscriber, you can contact me and I can set you up with instructions to get an account on our server.
You do NOT need to be a part of the Eggware Social server in order to join the Eggware Social space or its rooms.
Technical alert! On registration (detailed below), be sure to put the homeserver's full name in. The homeserver's name and its display name are not necessarily always the same. Generally, this is just dropping a subdomain for brevity's sake. For example, Eggware's homeserver is chat.eggware.social, but in your username it will display as "@username:eggware.social".
Clients
There are many clients and all of them have unique problems. I recommend Cinny, which is mostly feature-complete. For mobile, Fluffychat is alright, though it struggles with threads; Cinny also works just fine through mobile browser.
I don't recommend Element; you can't upload custom emoji and I also just don't like it very much. But if you really don't like Cinny and you have free time to waste, you can try out other clients. It might be a good idea to have a feature-complete test room where you can view how it handles threads, custom emojis, etc. You're probably going to find a client that works great except for one feature that you need. Get used to that.
Encryption
If you're just getting started and you're not really tech-y, I really don't think you need to worry about this too much. If you want to read encrypted rooms, make sure not to lose your recovery key, or you'll lose all of the encrypted messages you had access to before.
Why not just use Signal?
I'm adding this now because I've seen this being pitched a lot.
Signal is good for encrypted private conversations. If you want to use it, go for it! Really!
Signal is primarily a messenger. It's just not really a good "chat room" software and I wouldn't be recommending it to people looking to get off of Discord. Accounts need a phone number, which is a significant hurdle. Signal is open-source, but it's extremely centralized, so you don't get client options. It is also all hosted on a main server, so if Signal pulls the rug at any point, you're fucked.
I host Matrix because it has room for fun features, like bots that I can run without needing throwaway phone numbers. I also believe in its commitment to freedom of self-expression, something that other chat software seems to lack. If you're primarily concerned about mass-distributing warez, you... should have been on Signal in the first place.
Signal also has a reputation, which lead to it getting banned in Turkey, and it likely will be targeted in other countries if a mass exodus happens onto it. I just think that's worth keeping in mind and I rarely see it brought up.
One last thing: I know this software can be difficult. If you need my help with something, please don't take your anger out on me. I am not here making these choices to torture you!! I want it to be usable just as much as you do!!
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