Each instance is its own server/website but the servers talk to each other.
People and companies can both host their own instances though I am not aware of any servers owned by businesses.

If someone shut down an instance all the posts that have already been federated will still be on the other servers that instance has connected to but there will be no new posts.

The Fediverse in general does have instances owned by companies like how Mozzilla has its own Mastodon server.
I also think the Owners of wordpress also showed interest in the fediverse though I am not sure if they went anywhere with that.

This comment is an example of federation since I am from kbin.social but you are from lemmy.world yet I can see your post and comment on it.