MMOs are always best experienced with friends, so while a good majority of the non-instanced content is perfectly suited for solo players, I definitely recommend having friends along for the ride. In any case, the cooperative-specific content in WoW comes in three flavors:
Storyline quests in the world often culminate in some sort of large-scale battle with a powerful opponent, or defending a location against waves of enemies. These quests will be flagged as a Group quest in your log, and they’re designed for 2-5 players. Early on in the game, these will be simple affairs designed for only a couple players, where the point will simply be to kill an enemy more powerful than a solo player can handle, but once you’re further along, they’ll start requiring players to tank/heal/interrupt spellcasting like you might have to in a dungeon or raid.
The bread and butter of WoW’s co-op are the various five-man dungeons spread throughout the world. Players will need to assemble a well-balanced group to tackle these using the MMO “Holy Trinity” as a starting point: a Tank to keep enemies under control and soak damage (an appropriately specced Warrior/Paladin/Druid/Death Knight), a healer to keep everyone alive and manage debuffs cast on players (Priest/Shaman/Druid/Paladin), and an assortment of DPS (damage per second) classes (all classes have viable damage specs, but the pure DPS classes are Rogue/Mage/Warlock/Hunter) to well, stab enemies, shoot them or blow them up with magic.
Most dungeons consist of trash pulls (many groups of more powerful regular enemies who will need to be tanked and crowd controlled to defeat) and bosses (big baddies who have a ton of hit points and nasty abilities which will need to be anticipated and dealt with). Bosses are your best source of delicious, delicious loot, and players will often want to run dungeons to make sure they have the best gear.
Once you’ve reached maximum level (relative to each expansion), heroic versions of each dungeon become available. These re-tune all of the content to the abilities and stats of a max-level player, and quite often change the mechanics of boss encounters to be far more challenging. Bosses will have additional abilities, use them more often, and under-geared/unprepared players will have a much more difficult time defeating them.
Usefully, WoW includes a “Dungeon Finder”, which allows your group to fill out any remaining roles you might be missing and helpfully teleport you to the chosen dungeon when your group is filled out. You can also elect to do a random dungeon, which offers extra rewards as incentive once you reach level 70.
In addition to seven new level 80+ dungeons, Cataclysm adds a heroic mode to two perennial favorites of Classic WoW: Shadowfang Keep and the Deadmines.