Matrix isn't an app, it's a protocol. This means that Matrix isn't controlled by a single corporation or a single entity. Anyone can set up their own server that runs on the Matrix protocol. Anyone can make their own app that uses Matrix to talk to other people, and lots of people have! (We'll talk about the different options for Matrix chat clients later.) This makes Matrix much more resistant to the sort of enshittification that has plagued a lot of more centralized apps.

At its core, Matrix is just a set of programming rules that everyone who uses Matrix to talk to each other agrees to follow. Think of it like email. If your email account ends in @gmail.com, you can still send mail to people with @protonmail.com or @hotmail.com addresses, because even though you have accounts with different sites, all emails run on the same underlying protocol. Likewise, whether you're using Microsoft Outlook, the Mail app on an iPhone, or the Gmail website, you can use the same address to send emails. What server you're registered on and what app you're using to connect doesn't affect your ability to access the core network.

Yes. Any given client can talk to any other client. You can even use multiple different clients on the same device at the same time, if you want.

Not all clients are created equal! 
Some have features the others don't, and some are so basic they're just text you type into a command line. 
One of the best and worst things about Matrix is everyone can pick the client that works best for them.

Generally, no. Matrix supports bridging, which allows connection to other chat protocols, but most of them operate server-side, which requires you to be on a homeserver that operates a bridge (or be an admin of your own homeserver) in order to use one. A complete list of Matrix bridges is available here.

If you're looking for a single chat client that can support a variety of different protocols, we recommend Thunderbird or Pidgin. Thunderbird supports Matrix out of the box, while Pidgin can support it through a third-party plugin.

These general-purpose chat apps may not support some fancier Matrix-specific features, like voice and video calling.