Levels are an indication of relative power that can apply to characters, mobs, npcs, quests and items. The higher the level, the more powerful the thing related to it is.
Characters starts the game at level 1, and earn levels by gaining experience. The experience meter at the bottom of the screen will fill as the player accumulates experience, and the player will gain a level when the bar completes. Access to a higher level adds Health, Mana (for classes that have it), some attribute bonuses, and access to more powerful abilities and items.
The "level cap" is the highest level that players can presently reach. At WoW's initial release, the level cap was level 60, and was increased to 70 with The Burning Crusade, and 80 with Wrath of The Lich King.
NPCs' and Mobs' levels are constant. The level of an NPC indicates how powerful that target is in combat. Two mobs of the same level should pose a similar challenge, and award similar experience and rewards for being slain. There are also some special formulas regarding NPC levels in combat, such as chance to hit/resist. See Mechanics for more details.
Boss monsters are always considered 3 levels above your own, regardless of what your level is. The boss does not get more or less powerful as you level, and only affects combat mechanics.
Quests also have a level that they are designed for. While you do not have to be the intended level exactly to do a quest, you might not be able to acquire a quest if you are too low, or it might be considered trivial if you are too far above it.
Items have an 'Item level' (or iLevel) that determine how many points worth of bonuses they can have. There is no way to explicitly see an item's iLevel within the game -- you will either need to use a mod or look on a site such as this one. See Itemization Formulas for details on how item level affects an item.
When viewing another player, mob, npc, or even a quest in your quest log, you will see a color scheme to show you the difference in level between your character and the targeted 'object'.