If you're offline when your game would normally assign you an achievement, what happens? Are you notified, and it syncs to your profile later when you are online? Do you just miss out on it?
You're in luck: the Xbox Engineering team posted about this just today: Achievements Unlocked.
The basic idea is that the achievement is added to your local profile, which is marked as dirty. The next time that you connect to LIVE with that profile, it syncs up with the copy of your profile on Microsoft's servers.