Talent

A talent is an additional class-specific ability or power that requires talent points to gain or improve. Each class has three Talent Trees to choose from.

Basics

Personalizing your character is what separates each character of the same class. During the early Beta Test, the character Talent system was very rough and Blizzard went back to the drawing board to make the system more intuitive by adding prerequisites to the Talent system. If you have played the Diablo series of games you'll be more than familiar with the idea of a skill tree and this is the new system Blizzard has employed for World of Warcraft's Talent point assignment.

Why have Talents?

The reason talents are included is so that each player of the same class will be slightly different as far as how their skills work and how effective they are in certain areas. Certain talents are beneficial for leveling, while others help you do better in PvP, and others still can help you with Raiding. Without this system, each class character would be identical, which would remove some of the variety of the gameplay. It's up to the player to make his character different by specializing in the character's different Talents.

How does the Talent system work?

Talents are unlocked at level 10 and you begin receiving Talent Points. Level 10 grants you a Talent Point as well as level 11 and every level thereafter. Some classes have multiple roles depending on which tree you choose, such as Tank, DPS, or Healing, but some classes only have different playstyles of the same role; for example the Rogue is a DPS only class, so all his trees are DPS. The maximum amount of Talent Points is 51, though you can place points into any of the trees as you see fit, though the best talents are accessible only through specialization. Although this seems pretty straightforward, careful selection of talents and point placement can easily make the difference between one who opts for PvP as opposed to another who opts for PvE.

Some Talents are available immediately, and the rest are greyed out. This is because you first need to assign points to the first tier of talents (the top row) before proceeding to the next tier of talents. In Tier 1 you can begin placing Talent Points as soon as you reach level 10, Tier 2 requires 5 points in that tree in order to unlock, and each subsequent tier requires an extra 5 points in the tree (not necessarily in a certain tier) before being able to put points in the next tier of talents. The 7th tier of talents (the bottom row) typically has only 1 talent and is usually considered an "Ultimate" talent for that tree and are almost always very useful.

Once you have assigned a point into a talent, it is now allocated. Allocation has been finalized, you cannot retract that point unless you go to your class trainer and ask to relearn your talents. This will reset all of your talents in a tree allowing you to reallocate all of the Talent Points you've earned, but at a cost that increases each time you relearn your talents, so please choose wisely. This cost begins at 1 gold, but quickly escalates to 2, then 5, then 10 all the way up to 50. Talent costs above 10 gold slowly decay over time until they return to 10 gold.

Where can I find the talents?

There are a couple of ways to activate your Talents window. One method is to click the Talent icon on the Micro Menu of the interface bar at the bottom of the screen (assuming you are using the default UI). This pops up the Talent interface which features the main Talent window showing the tree and Talents. It also features three tabs along the bottom which can be clicked to flip between each of the three trees. You can also activate the Talent interface by pressing the N key (if you haven't unbound it or bound the key to a different action).

What if I choose the wrong talents?

Don't panic! Players have the ability to unlearn all their talents and start over. Your Class Trainer, for a modest fee (increases by 5 each use, to a maximum 50, however the first time you do this you will only be charged 1, a freebie of sorts), will let you recover all Talent Points spent up to this point. You can then reassign all your points. Keep in mind that all points are removed from ALL trees, and you'll also unlearn higher ranks of the spells achieved through a talent point expenditure.

You can avoid accidentally clicking the wrong talent by using the Kalented add-on, but otherwise careful planning and clicking is advised.

Since patch 1.11 there is a decay in the cost of such talent respecs:

The cost to unlearn talents will now decay over time. This cost will be reduced by a rate of 5 gold per month to a minimum of 10 gold.

Note: Although the official patch notes continue to say that the minimum is 10, subsequent official posts on the WoW forums have confirmed that that is a mistake, and the cost will only be reduced to a minimum of 15.

Advice on picking your talents

Pick one thing to do and stick with it. If you try to (for example) get a bit of healing ability and a bit of tanking ability, you'll probably end up bad at both.
This does NOT mean you should definitely put all your points in one tree. Once you get the "ultimate talent" at 31 points, look at your other trees. Consider this a very strong, well-founded suggestion as the base talents in many trees work well with the advanced talents in other trees. That's not to say, however, that a Mage who puts all 41 points into Frost is automatically an easy target in PvP. In the hands of a skilled player, a 41-point single tree build can be very effective.
Any ability that activates "whenever you are critically hit" is a PvP talent (PvE players won't get critically hit much) and can be safely skipped if you aren't doing much PvP.
Any ability that gradually applies several stacks of a buff or debuff is designed for boss fights, which last long enough for these abilities to reach a full stack. When you're leveling, your fights usually won't last long enough, so skip these talents if there's something better available.
Any talent that decreases threat caused by your abilities is obviously a PvE talent, designed for group situations such as instances. They are not recommended whatsoever for PvP play, and are completely unnecessary in that area.
Any talent that makes you go faster in terms of movement (or mount) speed will save you a tremendous amount of time and is almost always worth getting while leveling.
When leveling on a PvP realm, it is entirely understandable if you choose to create a versatile spec that allows you to perform decently enough in PvE but also gives you a boost in PvP capability in cases where you have to defend yourself from players of the opposing faction. In this case, this is a healthy alternative to focusing your talent build entirely on PvE or PvP.
After purchasing Dual Talent Specialization, most players usually opt to have one raiding build and one PvP build.
There's no such thing as a bad spec (except in very few cases). That's still no excuse to bring your PvP spec into an end-game raid, however. Please be aware of the strengths (and purpose) of your talent build, and utilize them accordingly.
If you see someone else saying that only certain talent builds are acceptable, pay no attention to him/her. Talents were implemented so that players could further customize their characters. Five Mages from the same guild having the exact same talent build because they were told to is not customization. Choose the talents that work best with your play-style, but do be mindful of which talents you end up choosing. Again, choose the talents that work best with your play-style and you will become stronger.
If you're having problems being accepted due to your talent build (which shouldn't happen, but sometimes it can), purchase Dual Talent Specialization. Now you can have both a cookie-cutter build and your own customized build, and you can switch between them any time you wish. Problem solved!