How do I refresh / check new mail in Mail?

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)

Or you can just edit your settings to get all the mail every minute, 5 minutes, 10 minutes, 30 minutes,...
It's under preferences - general. I've got it on 1 minute, so I never have to hit the button Diago described :)

...or you can go to the mailbox menu and "synchronize all accounts."

Also, holding down the Control key while pushing any of the buttons on the button bar will allow you to quickly change to "icon and text" to show what the buttons do, then quickly turn text off once you're clear.