In D&D, the words "adventure" and "quest" are virtually synonymous. They both mean a journey, fraught with danger that you undertake for a specific purpose. We sometimes joke that the game is all about killing monsters and taking their stuff, but the reality is that the game is about adventures. You go into the dungeon and kill monsters with a larger purpose in mind: to stop their raids on caravans, to rescue the townsfolk they've captured, to retrieve the lost Scepter of the Adamantine Kings for the rightful descendant of those kings.
Quests are the story glue that binds encounters together into adventures. They turn what would otherwise be a disjointed series of combats and interactions into a narrative -- a story with a beginning, a middle, and a climactic ending. They give characters a reason for doing what they do, and a feeling of accomplishment when they achieve their goals.
Quests can be major or minor, they can involve the whole group or just a single character's personal goals, and they have levels just like encounters do. Completing a quest always brings a reward in experience points (equal to an encounter of its level for a major quest, or a monster of its level for a minor quest), and it often brings monetary rewards as well (on par with its XP reward, balanced with the rest of the treasure in the adventure). They can also bring other rewards, of course -- grants of land or title, the promise of a future favor, and so on.
The idea of quest rewards is nothing new to D&D. Second Edition, in particular, promoted the idea of giving story rewards of experience points when players completed adventures. The quest rules in 4th Edition are directly descended from that idea, integrated into the economy of rewards in the game. They're a rules wrapper around the story of the game, a way to keep players mindful of the purposes behind all their adventuring.
One of the suggestions in the 4th Edition Dungeon Master's Guide is to give players a visual, tactile representation of a quest as soon as they begin it. At the start of the adventure, after the baron has briefed the characters on their mission and been bullied into paying them more than he intended, you can hand the players an index card spelling out the details of the quest -- including the agreed-upon reward. In the middle of the adventure, when the characters find a key with a ruby set in its bow, you can hand them a card, telling them that finding the matching lock is a quest.
When the players have cards or some other visual representation of their quests, it's easy for them to remember what they're supposed to be doing -- and to sort out goals that might be contradictory. That's a really interesting ramification of the quest system: It's okay to give the players quests they don't complete, quests that conflict with each other, or quests that conflict with the characters' alignments and values.
For example, the mentor of the group's paladin might ask him to find and destroy the Ruby Tome of Savrith the Undying. At the same time, a shady character is offering the rogue a sizable sum in exchange for the same tome, and the wizard's research turns up a reference to a ritual contained in the Ruby Tome that the characters will need to use in order to complete another quest. Three quests stand at odds, and it's up to the players to decide what they want to do.
There's a story that's a lot richer and more interesting than simply going into the dungeon to see what treasure is there.
