In short, yes, your characters do level up (armor doesn't). Your characters level up through experience (primarily gained in combat). Your ability to wear certain armor and equip weapons may or may not be contingent upon your current level. (The higher the value the more likely.) Moreover, the gear that you possess has (at least) four stats as well as their own "perks", which affect your character's own level and abilities/powers. In regards to level, there is a soft cap of 20 and a "hard cap" of 30, which you can attain by raising your Light stat (contingent upon the armor you're wearing) after level 20. Their are also [url=http://www.destinypedia.com/Reputation]"reputation ranks"[/url] that you can earn with vendors (the means by which you do this will vary) to acquire certain things or ability to purchase higher value items with them.