Automatically download messages
You can set up Thunderbird or Mozilla Suite to check for new messages when the application starts up, as well as at regular intervals (such as once every 10 minutes). To do so, go to "Tools -> Account Settings -> Server Settings" and then check the appropriate boxes for "Check for new messages at startup" and/or "Check for new messages every __ minutes." (In Mozilla Suite, access these settings via "Edit -> Mail & Newsgroup Account Settings -> Server Settings".)
For POP3 mail accounts, Thunderbird will not automatically download new messages for these settings unless you also select the box for "Automatically download new messages." If Thunderbird pops up an alert saying that you have one or more new messages but no new messages appear in your Inbox, it is likely because you have not selected the "Automatically download new messages" option.
Other related settings are under "Tools -> Account Settings -> Server Settings -> Advanced". Under "When downloading pop mail for this server, use the following folder for new mail:" you may have to select "Global Inbox" (Local Folders Account). By default this is set to "Inbox for this server's account", but if this folder does not exist it will not download new mail or explain the problem. When you create a new account, it will not create this folder for you.
If you check "Include this server when getting new mail", it will download mail when you click the "Get Mail" button or select File -> Get New Messages For -> Get All New Messages. If this is NOT checked, you must select the account from the "Get Mail" drop-down, or File -> Get New Messages For list.
If you have more than one email account, the above settings need to be set separately for each account.