Alchemy is a crafting skill and takes up one of your two primary tradeskill slots when learnt. It is used to create potions and elixirs to help your character in the game. Many of these potions and elixirs are popular with lots of players because of the effects their characters receive when drinking them. Furthermore some are also necessary in order to complete quests or in order to make items in other trades - a tailor will for example need 4  Healing Potions to make  Truefaith Gloves.

Another thing that alchemists can do are transmutes. This is where the alchemist turns one thing into another. An example of an old, popular transmute which alchemist could make money from was the arcanite transmute. With The Burning Crusade there are much more profitable transmutes out there though. Popular endgame transmutes are  Recipe: Transmute Earthstorm Diamond,  Recipe: Transmute Skyfire Diamond and  Primal Might. The only downside to transmutes is that they have a cooldown timer - usually this is 23 hours.

You can of course have any other primary trade paired up with Alchemy, but by far the best is Herbalism. With Herbalism you can pick your own herbs and then make them into end products. This is advantageous in multiple ways. First you won't be a slave to the auction house market. Prices can fluctuate and sometimes the herbs you want won't even be up for sale, which can be very frustrating. Also, combining Herbalism with Alchemy means that you don't have to spend money at the auction house for ingredients but can just go out and farm them yourself to maximise profits.

If you have more than one high level character it doesn’t have to be on the same character you have both Alchemy and Herbalism - as long as you have someone to go farm on when the auction house dries up. A personal example is that I have a character that is able to stealth, which I use for farming herbs and ores. I don’t use her in any endgame content, so I find it better to have two crafting professions on the ones I do use for endgame content. This is so they get the maximum benefit from the trade specific bind on pickup items.

I also warmly recommend that you take up Fishing on the side. It goes very nicely with Alchemy and you can get some very profitable skillups from being a fisherman as well.