Nintendo has released a video demonstrating how character progression will work in Fire Emblem Awakening, with plenty of gameplay footage from the upcoming 3DS title.
As far as levelling up goes, it's fairly standard. Characters gain experience in battle, and when they gain enough they go up a level. Going up a level means characters get a boost to their stats like HP and strength. Using a certain kind of weapon also levels that character up for that particular weapon type. The narrator for the video suggests it might be a good idea to use weaker characters in battle so that they too can level up and avoid being such easy targets for the enemy. So far, so obvious.
But there are also classes in the game, two types: base class and advanced class. An item called a “Master Seal” allows you to elevate your character from base class to advanced class, which substantially improves his or her stats. There are multiple advanced classes too, so a knight can become either a “general” or a “great knight”. Each has its own benefits, so you have to think tactically. A "Second Seal" will allow you to change between the multiple base classes, which has its own advantages, and once you get to advanced class you can switch between those (or back down to base class) as often as you'd like.
Fire Emblem Awakening was released last year in Japan, but will launch on February 4th of this year in North America and in April in Europe.