In the IMAP protocol, all E-Mails are stored and managed on the server. This allows multiple clients on different computers to manage the same mailbox.
By default, the Mail.app in OSX Lion also stores Drafts, Notes and Sent Messages on the server.
Just setup your mail account in System Preferences → Mail, Contacts & Calendars and you should be fine. Your E-Mails will be downloaded from the server.
Best Answer
Apple Mail's check for mail icon looks like a new mail message icon, and can be found here:
If it can't be found you can add by right-clicking on the toolbar and choosing customize:
You can then select the icon from the list and drag it to the toolbar at the top:
You can also use "Mailbox" -> "Get All New Mail" which has a shortcut ⌘ + ⇧ + N.
You don't need to synchronize. Synchronize does more than a new mail check (see this answer)