Welcome to Lemmy/the Fediverse! It's 95% Reddit ant 5% email.
What is a server?
A server is just a computer that runs software that is always standing ready to do something for a user. For the purposes of our discussion, a server is the computer that runs an instance of Lemmy.
Let's start with something we're probably all familiar with: Reddit. The way Reddit works is, it's a big company that owns (or rents) a bunch of servers. All of the data anyone has ever posted to Reddit, plus the software that makes Reddit go, is stored on these servers. Users open an account with Reddit, and they're given access to these servers. Reddit controls ALL of them so it doesn't matter much exactly which server you attach to. They might all be in one location, they might be scattered all over the world, it's fairly transparent to users. The various websites or apps we (used to) use to connect to Reddit were simply front-ends which allowed us to access the server.
Lemmy is software that does a lot of what Reddit does. It's software you run on your server which keeps track of users, communities, posts and comments. Unlike Reddit, which keeps its "back end" software a closely guarded secret, Lemmy is open source. Anyone can read the source code or run it on their own server.
What is an instance?
When someone runs a copy of Lemmy on their own server, that is an "instance." Each Lemmy instance is its own little Reddit. You can sign up for an account on an instance, and the experience is very much like Reddit; "subreddits" are called "communities" here. The owner of the server is the admin, individual members can make posts to communities, comment on those posts, create new communities and moderate them...it's its own little Reddit. lemmy.world is its own little reddit, sh.itjust.works is its own little reddit. Well what good does a bunch of little Reddits do?
What is a federation?
A federation is a group of independent organizations that work together as a group. Think of the United States of America: 50 states that each have their own constitutions, codes of law, legislatures, executives, courts etc. which are united under one common Federal government. Lemmy works this way; each instance is a member of the Fediverse, a federation of servers that run software which uses the ActivityPub protocol--a standardized way for servers to transfer various kinds of social media data between them.
Every instance of Lemmy is its own website, run on its own hardware, with its own communities, members, admins and such. Each instance will have its own house rules; some allow political discussions, some don't. Some allow pornography, some don't. Some restrict community topics to a broad field, like music or video games. Some are general purpose instances. It's up to the admin of each instance. But, each instance is able to communicate with all of the others, so having an account on one instance allows you to communicate with all the others--with some caveats I'll talk about later.
In fact, the ActivityPub protocol is used by other services besides Lemmy--another similar service is called Kbin, which is functionally similar but the software is written by different people. Lemmy instances can communicate with Kbin instances and vice versa.
So where every instance is its own little Reddit, all of the instances kinda voltron together to make one big Mega Reddit.