Apple's Mail e-mail client is a fairly straightforward tool for accessing IMAP, POP, and Exchange accounts, and for the most part it works quite well, but it is not without its quirks. One of these is that at times Mail can start prompting users for their account passwords again and seem to not properly store them. When this happens the program may appear to stop interacting with the mail server for such functions as downloading new messages.
The first thing to do whenever there is a problem with Mail's connection with an e-mail server is to open the Activity window and try manually invoking the connections. To do this, go to the Window menu and choose Activity (or press Command-0) and then click the Get Mail option in the Mail toolbar.
This should have Mail make a full connection and synchronization attempt with the mail server, and you should see various progress bars for this activity in the Activity window. Sometimes connections can take what seems like a long time, especially while attachments are being cached and synchronization between mailboxes is being done. Therefore, be sure to wait for all progress bars to complete in the Activity window before continuing to troubleshoot. If a connection seems to pause with no progress, you can click the red stop-sign button next to it to cancel that task.
The next step is to clear your stored passwords for Mail so you can enter them again. Do this by opening the Keychain Access utility and selecting the "login" keychain. Then search for any entries associated with your e-mail account (such as "gmail" or "aol"). The password for these should be an Internet Password type of keychain entry with a blue @ symbol as an icon. To check the entry's association with Mail, double-click it and select the Access Control tab. In the list of applications you should see Mail, in which case close the window and remove the entry from your keychain.
After all keychain entries for the problematic mail account have been removed, launch Mail; if the program does not automatically check for new mail, click the Get Mail button. You should be prompted for your account password, and when you supply it be sure to check the option to save the password in your keychain.
When you supply your password again, be sure the Activity window is open and you should see the program access your account. Sometimes if account synchronization takes a while or pauses then you may not see any changes in the Mail application window, which can cause people to assume the newly supplied password did not work.
After all synchronization, caching, and other activities have finished in the Activity window, send yourself an e-mail to test the account and make sure everything is working again.
Beyond account passwords, sometimes various problems can happen with other Mail settings, so if Mail appears to stall even after refreshing the password, you might try some maintenance routines on various mailboxes. Go to each of your mailboxes (especially your inboxes) and with each one selected choose Rebuild from the Mailbox menu. In addition, try cleaning up your mailboxes by removing or archiving older messages and deleting those that have attachments.