I'll suppose you played Warcraft 3 and it's expansion.
World of Warcraft is an open-world MMORPG, the most popular MMORPG in the world. With your account you can make up to 10 characters per server and 50 in total. Characters have different classes and races, which you pick when you create them. Classes are warrior, paladin, hunter, shaman, rogue, druid, priest, mage, and warlock. The death knight hero class (starts at level 55) is added for those who have Wrath of the Lich King expansion and have a character leveled to 55.
There are 8 races, 4 for the alliance and 4 for the horde. Two races are added for those who have the Burning Crusade expansion. The horde and the alliance cannot speak, trade, or group with each other, but they can kill each other. Most Player vs. player combat is between the horde and the alliance. The alliance races are human, dwarf, gnome, night elf, and draenei (burning crusade). The horde races are orc, troll, tauren, undead, and blood elf (burning crusade).
The character levels through questing, killing, and other ways until he reaches the level cap. The level cap is 60. 70 for those with the Burning Crusade expansion. 80 for those with Wrath of the Lich king expansion. Wrath of the lich king cannot be implemented without the Burning Crusade.
Story:
World of Warcraft starts 5 years after the battle of mount hyjal. Every few months a patch is added that pushes the story further and adds new things. Expansions push the story a lot more. The horde and the alliance are at constant war with each other. The horde, lead by Thrall, consists mainly of orcs, darkspear trolls, tauren, and the undead forsaken lead by Sylvanas Windrunner. The horde consists mainly of the humans of Stormwind, the dwarves of Khaz Modan, gnomes, and night elves. Your character adventures through Azeroth, with many stories in this world.
-The Burning Crusade starts in patch 2.1. Before it's release, an in-game event was made. The dark portal was reopened, with hundreds of demons marching to Azeroth. High level players are requested to fight. When the Burning Crusade is released, anyone who buys it can go to Draenor (now called Outland) through the dark portal. The blood elves joined forces with the horde, thanks to the fact that Sylvanas was one of them when she wasn't an undead. A space ship crashed on an island near night elf settlements. Inside the ship, there were the Draenei race that came from Draenor. The draenei joined the alliance. The story develops from patch 2.1 until patch 2.4.
-Wrath of the Lich King starts in patch 3.1. The event before its release is boxes spreading diseases in the main cities. Players turn into undead (temporarily). This is the first attack. When this attack is beaten the second event comes. Tens of undead abominations, dragons, ghouls, and other stuff attack Orgrimmar and Stormwind. The alliance and the horde decide that they should march to Northrend. The story develops and is still developing.
The story is much more to that. It's just that I'll need to write a book if I wanted to tell you a real summary of it.
The guy who said that there isn't much to the story doesn't know the true essence of the game. The story is one of the best games stories I've seen.
If you want to know more about the story you can go to WoW.com and read know your lore articles.
Don't read the story in the official site. It hasn't been updated for years.