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.
Thank you very much for your support. As I was not able to receive emails I didn't access your email helping me to sort it out. After hours and hours go round in circles, I went to an Apple repair technician and he sorted it this morning. Then I received your email. Thanks for letting me know to always use Cancel for a mail.app password Box. Thanks for advising me about Snow Leopard which seems a good idea. Would that enable me to access ICloud?
In no particular order...
There's no need to use mail with the forums; I don't. You can view your own communities postings via the web interface.
10.6.8 gets you access to the Mac App Store, and to some other features, and it's the path to getting newer software and apps loaded onto your system, and for what your hardware will support. iCloud isn't particularly supported until 10.7 and — for various stuff — 10.8. You'll have access to the iCloud web interfaces and to iCloud mail on 10.6 and earlier, but not much beyond that.
Minimally OS X 10.7 and increasingly the current OS X 10.8 release are required for various applications. Apple doesn't generally keep support or bug fixes for older versions around, and never really has. Just the current and immediately-previous version get patches and updates, mostly.
OS X 10.6.8 is the minimum release for a few key software packages, such as iTunes. iTunes 10.6.3 is the last version with support for OS X 10.5, and that version of iTunes is now too old for iOS 6 devices, for instance. You need iTunes 10.7 or later (and OS X 10.6.8 or later) for iOS 6 devices.
Here are the minimum system requirements for 10.6 and 10.7. Given you're presently running the very-old OS X 10.5, your Mac hardware likely won't meet the OS X 10.8 system requirements, and might not even meet those of OS X 10.6 or 10.7.