Most of the class summaries you find around the web are ... dated. They were written many patches ago. Since then, classes have changed radically. Even without the obvious example of the shaman's once being able to tank, a lot of how the classes feel when you play them is radically different from when the game started. Consider this a refresher.
If you're truly new to WoW, there are a few terms you should understand in this guide. There are three generally accepted roles when you group up to do dungeons and raids. Tanks are the big, tough guys who keep the enemy focused on themselves, because a tank is capable of taking the damage. Healers repair the damage taken by tanks and the rest of the party. DPS stands for damage per second, and it's become shorthand for all the folks who lay the smack down on the enemy. While all characters can do damage, DPSers can do multiplicatively more damage than anyone else.
Crowd control allows you to briefly keep an enemy from fighting you. For example, Polymorph turns an enemy into a sheep. Enemies can't hurt anyone if they're sheep, right?
Role Damage and tank
Crowd control No "pure" crowd control options, but a lot of snares and slows and such
Death knights were introduced well after the release of WoW. They use runes to empower their long list of damaging abilities. Death knights are one of the most powerful and effective damage and tanking classes in the game, mostly because they don't have obvious class flaws. Death knights are competent against ranged and melee characters alike. Their signature Death Grip ability even allows them to force an enemy to fight on the death knight's terms. In order to play a death knight, however, you must already have another character to level 55.
Role Damage, heal, tank
Crowd control Only against certain types of enemies, plus a Root
Druids are the most versatile class in the game. They can be tanks and healers, but they can also do damage from either close up or a distance. This makes druids an excellent option for someone who wants to fit into any group. However, since druids do have so much going on, they can be difficult to truly master. Druids have a smattering of buffs, debuffs, and crowd control, as well, which means they generally help out any group they're in.
Crowd control Selective traps, sundry "sleeping" shots
Hunters stand back and shoot their enemies with bows and guns. The hunter class was once considered the easiest class to play, but that has significantly changed in the last two expansions. Hunters have a huge array of tools for any group to use. What makes hunters special is that they get a constant companion pet, which they choose by going out into the wild and taming. Each different type of pet has different abilities of its own, making hunters a surprisingly deep class to play.
Crowd control Masters of crowd control, including Polymorphs, roots, snares, slows, and silences
The mage class is the traditional "wizard" class of WoW. They do damage at a distance by hurling fire, ice, and raw arcane energy. The mage class has many, many options for controlling enemies, including turning them into animals, trapping them in place, and slowing their movement. Mages use mana to fuel their attacks. Being a mage is a great option for new and old characters, since a mage tends to focus on a single thing and does that single thing very well: blowing up the bad guys.
Role Damage, heal, tank
Crowd control Only in one talent spec
Paladins are one of two classes capable of doing all three key dungeon roles. Unlike druids, however, paladins can only perform the "melee DPS" role. Paladins use mana to fuel their abilities and occasionally make use of Holy Power to use special powers. The high survivability of paladins make them a great class for soloing and for PvP. As such, though, you're likely to find many, many other paladins in dungeon groups, which makes it a little slower to get gear (because you have to share.)
Role Damage, heal
Crowd Control Limited options
Priests are the archetypical healer class. They summon the power of the Light and Shadow to smite their enemies and grant solace to their allies. Even in their damage-dealing configuration, priests do a lot of healing to their allies. Priests are almost always a welcome class in any guild and desperately sought after all over Azeroth.
Crowd control Sap
Rogues are the sneaky assassins and dashing swashbucklers. They use a sort of low-grade mysticism to stealth around and stab their enemies in the back. The rogue class is a pretty capable class for solo play, and rogues do a very high amount of damage. As with paladins, though, you tend to see many, many rogues around, which can make competing to get in groups kind of tough.
Role Damage, heal
Crowd control Turns enemies into frogs, can affect Elementals
The shaman class is a bit of a unique snowflake. They can do damage at a distance or in melee range, depending on their spec. What makes shaman special is their totems. They drop little totems on the ground that create buffs, heals, and other effects on behalf of the shaman. Managing totems isn't very difficult, but relying on them can feel like a drawback if you need to move around.
Crowd control Fears and pet-based
Warlocks sit at the distance and use unholy shadow and fire to blow up the bad guys. They work with a series of demonic pets to accomplish their task, although only one pet can usually be active at once. Warlocks are a strong damage class, but their button rotations and wide array of powers mean they aren't well-suited to brand new players. Once mastered, however, warlocks are known for their dominance of PvP and ability to put out compelling damage.
Roles: Damage, tank
Crowd control None
Warriors are the traditional "rawr, I hit the bad guy with weapons" class. They're based on a rage mechanic that doesn't really have an equivalent anywhere in traditional fantasy games; essentially, the more they deal or take damage, the more rage they get. They use this rage to fuel their attacks. For this reason, warriors can frequently feel "gimped" if they're not getting enough rage. Warriors have a reliable interrupt, like all melee, but lack any kind of crowd control. Warriors are a straightforward class that relies on mastery of their rotation to get the most effect.
Visit the WoW Rookie Guide for links to everything you need to get started as a new player, from how to control your character and camera angles when you're just starting out, to learning how to tank, getting up to speed for heroics and even how to win Tol Barad.