The OP asked about pop3-mail. That's copied to the local mail-storage (dbx in OE, pst in Outlook, and I still don't understand how Thunderbird does it) and deleted from the server (either immediately, either after some time, depending on the settings).
Of course, att.net only tells you about connecting to their servers (user name, password, pop3- and smpt server and maybe some other settings to enter into your pop3-account definition). They aren't interested at all in what you do with your mail after receiving it. So they don't say anything about backup and copying to new machines and to new email clients.
Webmail is just a way of looking into the mail stored at your mail-provider. With gmail and hotmail and such that's all your mail (unless you use pop3 also). With fully pop3-accounts it's usually only the new mails you didn't yet copy to your PC and delete from the server.
For webmail no setup at all is required. Just enter the right url (like gmail.com), then your userid and password and there you go.