A server isn't necessarily all that meaningfully different from Joe's laptop, it's just that Joe's laptop isn't exactly practical for running big things. My website runs on a real server, but it doesn't really act much different from a computer I have in my living room when I remotely log into it, I can access all the same files, run all the same software if I wanted to.

Federated is just jargon for "the posts and comments from here will display over there too because my computer knows yours exists and runs the same software and the software does the legwork of meshing all of that shit together with mine"

Yes, "instance" is the word used on Lemmy for what is usually called "server" elsewhere.