Alchemy

Alchemy is one of WoW's crafting professions - it requires materials produced with another profession, usually Herbalism. Alchemists turn various reagents (mostly Herbs) into powerful consumables that fall under one of the three categories:

Elixirs - they provide fairly powerful and versatile buffs over a long period of time, but their effects are lost if the player dies. Each player can have up to two elixirs active - one Guardian and one Battle Elixir. Common elixir effects are the ones that increase a player's secondary ratings - hit, expertise, mastery, etc.
Flasks - these provide relatively basic, but vital buffs over a very long period of time. Flasks' effects also persist through death, but only one can be active at any given time. Flasks increase primary stats - strength, agility, intellect, spirit, or stamina.
Potions - Often overlooked, potions provide an instantaneous or "bursty" buff. Because of their concentrated power, potions are limited to one per combat. Common effects include a massive burst in a primary stat, or an instantaneous replenishment of some health or mana.

Aside from combat-oriented brews, alchemists are also able to transmute elemental reagents, ores, and gems into their more powerful counterparts. For example, Pyrium Bars can be created either by smelting the rare Pyrite Ore or by having an alchemist transmute it from one Elementium Bar and one Volatile Earth, both of which are very common and cheap. Other transmutations include the turning of three uncommon gems and some herbs into a rare gem, as well as the change of one Volatile element (a much-needed reagent for many professions) into another.

Alchemy is not all about battle-readiness and helping your allies with their professions though - it can be fun and games too. Skilled and lucky alchemists who have decided to study Archaeology as well can craft a Vial of the Sands, which teaches its user the exotic Sandstone Drake mount. Potion of Treasure Finding can bring a little bit of extra income, and Deepstone Oil can, well... cause stiffness. Get you stoned. Literally.
Profession Perk
Regardless of their skill, all alchemists learn Mixology, which increases the effectiveness and duration of all concoctions they are able to make. This effect is personal, so only the alchemist will receive the extra effect. In addition, although not a strict part of the perk, alchemists of sufficiently high skill are able to create epic Philosopher's Stone trinkets that not only provide very good bonuses, but also increase the effects of healing and mana potions when used by the alchemist.