Although players may pick a class to select their particular skills, the world would be fairly bland if every single warrior were the same, every single druid, every single warlock. As a result, one of the aspects of developing your character involves selecting areas of focus.
Players focus their characters through the use of talents. Each class has three Talent Trees that specialize in one of the major aspects of the class. For some classes, these trees are very distinct. Druids, for example, have a healing tree, a spell damage tree, and a shapeshifting tree. Other classes are more subtle, such as warriors who have one tree for protection and tanking, and then two damage-based trees that differ in style.
In addition to the table below, you can find a Talent Calculator at WoWHead for inspecting each of the classes' talents.
Players begin earning talent points at level 10. With each level, other than 1-9, they will pick up 1 talent point. This means that, as of now while the level cap is 70, a player will earn a maximum of 61 points.
You can see the number of talent points you have unspent by opening the talent window. The default key for this is N. Also in the talent window are the three different trees, on separate tabs. Spending your talent points is as simple as clicking on the available talent you wish to add to your character. Be careful -- you can not simply unspend those points or withdraw them, so don't click by accident. Use the Talent Calculator to plan your decisions.
Talents, in general, give passive bonuses. Some of these bonuses affect the character as a whole, while others enhance specific abilities. For example, in the hunter beast mastery tree, the top two talents are Improved Aspect of the Hawk and Endurance Training. Improved Hawk gives a bonus to a particular ability -- Aspect of the Hawk. On the other hand, Endurance Training simply increases the health of the hunter and his pet.
It is possible to spend more than one talent point on an ability. Passive talents such as these offer a certain bonus for each point spent in that talent, and will normally accept 2, 3, or 5 points to multiply the effect. The first point in Endurance Training increases the pet's health by 2%, but completing all five points of the talent would increase the pet's health by 10%.
It is not necessary to only spend your talent points in a single talent tree, however as a general trend, the talents become more powerful or more specialized as the player goes deeper into the tree.
On every odd-numbered tier except the first one, there will be a single one-point ability. These are nearly always activated abilities that your character will gain by spending the point in that talent. Most of these are rather powerful, so it follows that they are usually the best single-point investments you will make in a talent tree. Players often design their talent build to collect certain key 1-point talents. Sometimes these abilities will have ranks that can be learned from your class trainer. This creates confusion from new players sometimes who wonder, "I see Aimed Shot (Rank 2), but where is Aimed Shot (Rank 1)?"
Not all talents can be acquired immediately. Every row of the talent tree, often referred to as atier, can not be accessed until an average of 5 points have been spent on each row above it. So for a third-tier talent like Unleashed Fury, (again in hunter beast mastery,) a total of 10 points would need to be spent in the first two rows. This does NOT mean that the player must spend 5 points in the second tier. 10 points all spent in the first row will make the third-tier talents available.
There are also sometimes specific restrictions. There is an arrow pointing from Ferocity (tier 4) to Frenzy (tier 6) that indicates that points must be spent in Ferocity before Frenzy can be accessed. The prerequisites always need to be maxed out completely to unlock the dependent talent. This is in addition to the normal point restrictions for that row, so you would need to have spent 5 points in Ferocity, and 20 points somewhere else in the Beast Mastery tree for a total of 25, to get Frenzy since it has 5 rows above it.
A few important observations about talent trees:
The specific assortment of points a player uses for their talents is often called their talent build or their spec, short for specialization or specification. Max-level characters can often spend a good bit of time concerned with the way their points are spent. For some classes, talent builds are simply a stylistic choice -- rogues, for example, have three different trees that deal damage in somewhat different ways, and although it affects their playstyle, it does not affect their combat role. For a hybrid class like a paladin or druid, however, the use of talent points often determines what the character will be effective at, and as their time in Azeroth continues, they may find themselves thrust into another role and wish to change their spec to match.
Fortunately, talent points are not completely permanent. When visiting a class trainer, they will offer you the chance to unlearn your talents. This will free up all of the points you have spent, but also costs money. The first respec costs 1 gold, and then it costs multiples of 5 gold after that to a maximum of 50. Choose wisely!