There are a total of 44 units that you can control in this game. Here is the list of their names in alphabetical order, click on their names to view their detailed information.
All units will stop gaining experience after they reach level 20, but if they are unpromoted, you can promote them to a new class starting at level 1 using specific promotion items. For a list of where you can find promotion items, click here. Any unpromoted units can be promoted after level 10(except Eliwood/Hector in their respective story mode).
Thanks to www.fireemblem.net for letting me use their info. All the stat growth, max stat, and promotion changes are taken from their site, thanks again.