Triage

Horde

Once you hit a character level of 40 and with at least 225 in your First Aid skill visit Arnok, in The Valley of Spirit, Orgrimmar who will give you the quest "Horde Trauma" which sends you off to see Doctor Gregory Victor in Hammerfall who will ask for you to assist him in healing patients in a quest called "Triage" which asks you to save the lives of 15 of his patients.

Alliance

Once you hit a character level of 40 and with at least 225 in your First Aid skill visit Nissa Firestone, in The Great Forge, Ironforge who will give you the quest "Alliance Trauma" which sends you off to see Doctor Gustaf VanHowzen in Theramore Isle, Dustwallow Marsh. Doctor VanHowzen will ask for you to assist him in healing patients in a quest called "Triage" which asks you to save the lives of 15 of his patients.
Both

Triage BandagesBoth doctors will instruct you to heal the critically injured first but it's advisable to let them die because unless you start healing them as soon as they spawn they'll usually die on you anyway and that's time wasted, time that you could have been healing a badly injured person instead. The Dr will give you 20 Triage bandages to perform the healing with. Below are some tips to completing this.

1. Press Ctrl V to activate the health bar over the heads of the patients for easy health monitoring.
2. Pull your camera out to third person so you can see as much of the room as possible. Tick "Show NPC names" in Interface Options and also press "V" on your keyboard.
3. Put the triage bandages the doctor gave you in your aciton bar. Do not use your own.
4. Position yourself in the middle of the room.
5. To see all around the room you can either hold down the left mouse button to swivel the camera around you or the right mouse button to actually swivel yourself around on the spot, whichever you find allows you to click on dying patients easier. In the Horde hospital you don't even need to swivel really.
6. When a patient appears, dying, click on the Triage bandage in your action bar to begin healing.
7. Keep your eye on other patients, swivel yourself or camera round as described above to monitor spawns all around you.
8. When you see another appear left click on them to pre-select them and as soon as your current heal has finished click the bandages in your action bar and the pre-selected one will begin to heal. Saves valuable seconds this way.

As the patients are healed they'll jump up, cheer and then run to the Dr. If you fail (6 patients die) throw any remaining Triage bandages away and re-request the quest from the Doctor, he'll give you a fresh batch of bandages.

I would just like to correct danyjr slightly.

This quest is available for horde when:
(1) you are level 35 (not 40)
(2) you have 225 First Aid

I'm sure this is the same for Alliance, but have not been able to check.

Also, this was verified in Patch 2.01.

A little notice:
Undead patient --> Critically Injured
Troll patient --> Badly Injured
Orc patient --> Injured