The migration from MobileMe and earlier systems over to Apple's iCloud all happened a while back, and most of the mail access moved over to iCloud and related servers.
First, confirm you can access your iCloud mail using your mail.mac.com username and password and the iCloud web interface. This will verify your account and your credentials.
Here are the set-up details (HT4864) for the iCloud SMTP server — the path used to send messages from the client to the mail server — and for IMAP — the path used to view messages stored on the server, and to receive new messages from the server. Verify your mail settings match these ports and servers, and that you have SSL/TLS enabled.
You must have both the receive path and the send path correctly specified and working, and these have separate connection set-up pages — the SMTP server set-up configuration page is buried — for Mail.app to work.
Once you have what you think is a functional configuration (or if you hit something that you can't figure out), use Window > Connection Doctor tool within Mail.app to check the access, and open up the details display to see what's happening with some diagnostics if you get a warning about any of your accounts.
In very rare cases, I've seen keychain entries get corrupted, so removing the keychain entries for both the IMAP and SMTP server passwords might be necessary, but that's very rare. It's far more common to have incorrect settings for the SMTP and IMAP servers, bad credentials — folks inevitably try different passwords in those pop-up password boxes, which causes the mail server to lock you out due to wrong-password input, which causes connection problems, etc — NEVER ENTER A PASSWORD INTO A MAIL.APP PASSWORD DIALOG BOX, IMHO THE CORRECT ANSWER IS ALWAYS CANCEL. Don't try different passwords. If it is a keychain corruption, see the Keychain Access tool from Applications > Utilities folder.
FWIW, if you are really on 10.5 as indicated by your footer and if your hardware supports it, then you may want to acquire 10.6 Snow Leopard from Apple. It's ~US$20. Here is the US Apple store for the Snow Leopard DVD purchase, and the specs and system requirements are posted there. This will get you to 10.6.8, which is a step up from what 10.5 offered.