The setup described here enables you to store all your email (and email for other people) on a single machine, which might be a home server, a remotely hosted server, or even a desktop, and then access it from anywhere.
The mail protocol most people are most familiar with is POP, which has long been the industry standard for serving and retrieving e-mail. A client, which is the sort of desktop mail program with which everyone's familiar, connects to the POP server and says, "Do you have any messages for me?" If the answer is yes, the client gets a list of the messages, downloads them, and optionally either deletes them from the server or leaves them in place. That's pretty much the entire capability of POP.