Fire Emblem is a strategy rpg. Did you ever play Advance Wars or Final Fantasy Tactics Advance on the GBA? It's pretty similar to that.
edit - IGN has a really great interview that talks about the GC Fire Emblem. Here's the link for that.
http://cube.ign.com/articles/654/654502p1.html
Basically you have a party and you're set on a playing field, and you have goals to accomplish. In Fire Emblem, there are 4 different kinds of goals.
1. Defeat the boss - To beat the stage, you have to defeat the boss. Usually you'll want to kill all the enemies for the XP, but you don't have to.
2. Seize the throne/building - Exactly what it sounds like. You have to seize the throne. This is another one where you don't have to kill all the enemies to get past the level.
3. Kill all enemies - Exactly what it sounds like.
4. Don't Die - This is also exactly what it sounds like. You'll be given a certain amount of turns that you have to live through in order to beat the level.
I've played Advance Wars, FFTA, and Fire Emblem, and Fire Emblem is definitely my favorite. Fire Emblem: Sacred Stones (for GBA), especially, is really awesome. It has two different storylines that you can play, and you only get the whole story by playing through both. It also has multiple class change options, so you have a lot of options with your characters. And unlike the first Fire Emblem for GBA, there's a lot of sidequest things that you can do to level up your characters.
Fire Emblem might be a little hardcore for some though, since if your character dies, they are gone forever. If you want to keep them, you have to start that level over again and try not to get that character killed.
I've put at least 40 hours into the GBA version, so I can't wait for the GC one, which should be even longer!
Make sure you check out the screenshots, too.