Questing is a vital element of World of Warcraft for giving the player a sense of purpose, especially while leveling. Quests also tie the players into the world by explaining elements of Warcraft lore, and can often give respectable rewards for completion.
Quests are very simple in essence: someone gives you a mission, you do it, and report back to them (or perhaps another appropriate person.) However, not every NPC in the game is a questgiver. Characters with a yellow exclamation point over their head will have a quest available to you. Right-click on that character and you will be able to read about the quest and choose whether or not to click on "Accept" and take the quest.
When you report back to the questgiver, you will need to click on "Continue". At that point, you will get some feedback from the questgiver -- usually a little bit of lore or a thank you -- and probably be offered a quest reward. You may have to choose between one of several options. If this is the case, you should probably try to find a reward appropriate for your class. Clicking on "Complete Quest" will finish the quest and give you the promised rewards as well as some experience and perhaps some reputation as well.
While Blizzard attempts to make the quests as interesting as possible, quests overall can generally be classified into one of a few different categories. Others have been attempted, but generally do not work very well.
Quests also exist for a variety of purposes, and while most quests are intended for the progression of one's character, they often have other motives:
Not every quest will immediately be available to a new level 1 character. Some quests have a required level before the questgiver will approach you. If you are near this minimum level, a questgiver may have a silver exclamation point over his head, as if to say "not yet, but soon." There will sometimes be other requirements for beginning a quest, however. Reputation sometimes plays a role in which quests are available.
The most frequent reason a quest is unavailable is because there is another quest that must be finished first. The quest Investigate Echo Ridge in Elwynn Forest is not available from Marshal McBride until Kobold Camp Cleanup has been finished first. Since these quests very frequently follow a storyline or otherwise have an "A leads to B, which leads to C" feeling to them, and often all come from the same person, players refer to these as Quest Chains.
Quest chains will often not have an item reward until the very end of the chain, and links in the chain cannot be shared with other players who have not done the quests leading up to it.
Most quests are designed with the single player in mind as a single mission. However, there are some quests that are handled differently.
Repeatable quests may be performed more than once. Sometimes these quests are infinitely repeatable, and sometimes they are only repeatable until a certain condition has been met such as achieving a certain reputation rank. Repeatable quests will normally not appear in your Quest Log after the first completion, and the questgiver will sometimes have a blue question mark over their head. Repeatable quests seldom have a significant reward, and are usually intended for raising reputation with a particular group.
A variation of the repeatable quest is the daily quest. These quests were introduced for level 70 players to more easily earn money, and are mostly found in certain high-level areas of Outland. Daily quests are repeatable, however they may only be completed once per day. They normally have a noticeable Gold reward. PvP quests allow the player to complete a PvP objective each day for Gold and Honor. These PvP Daily Quests may be started at level 31, providing the player can enter the targeted battleground.
Group Quests, formerly refered to as Elite quests, are too difficult for a single player to be expected to complete at the appropriate level. Group quests will often indicate how many players are recommended to finish them in the quest log. Group quests will be difficult either because they involve fending off large numbers of MoBs, or because they deal with killing Elite targets. Finding a party to complete a group quest can sometimes prove troublesome, but the rewards are often better than non-group quests.
Instance quests, marked "Dungeon" in your quest log, are more extreme versions of group quests that will take place in dungeons. After gaining Honored Reputation with certain groups, you may enter in the Heroic version of all Outlands instances.
Raid require a much more specialized and elite group of players. Normally when a player enters a raid, their quests may no longer be completed. A raid quest is a quest that will still be available while the character is part of, and usually requires, a raid. Raids can consist of as little as 10, or as many as 40 players.
The quest log will sort all of your quests by their related zones or events, then by relative difficulty, and will also indicate if a quest is daily, group, or to be completed in an instance or a raid. A character can have a maximum of 25 quests in their log, and will get an error message if they attempt to add a 26th. As a result, it is recommended that a player either completes old quests or abandons them.
The quest log will give feedback on each quest, including specific objectives, and the questgiver's explanation of the quest. It is often important to read this information to understand exactly how to perform the quest.
To assist in completing quests with other people, the quest log also offers a player the opportunity to share their quests with groupmates. To receive a quest, the player must be very near the sharing player, must be able to start the quest normally, must have room in their quest log, and cannot have other interface windows open. The most frequent reason a quest will be unsharable is if it is a later part of a chain that the player has not begun.