Assisted Power Leveling
Using any form of assistance such as:
High level priest/druids casting healing over time spells while out of group.
You can use any of your favorite classes as your focus character and plow through normal and high level quests much quicker and carelessly with a level 70 Druid casting Regrowth and Renew on you. You can use keyclone to avoid alt-tabbing. This can be an effective compliment to Mage/Paladin AoE grinding. Throw Thorns, Regrowth, Renew on a Paladin and drop consecration bombs into huge groups of mobs. The same can be accomplished to keep a Mage alive longer while aoe grinding.
Put a Paladin, Priest and Druid on /follow and you'll be the hottest new guy in the Barrens. Be warned, expect people asking you why you need 3 level 70s helping you level.
Using a high level character to do dungeon run throughs.
This is probably the most used method of power leveling, although a lot of times it is inefficient and has some drawbacks. I have power leveled at least a half dozen characters from 15 to 50 with this method, and a few into the mid-60s. The fastest I've used was an S2 Fury Warrior with a power leveling build. I've seen 70 BM Hunters clear the Cathedral in one pull (sending the pet straight for the boss), Afflication Warlocks clear SM Armory in sub-6 minutes and Paladins working their way clumsily through Cathedral with Consecration and bubble heals.
The best rundown works like this: Stockades 15-27, Scarlet Monetary until level 39, Zul'Farak until level 47. From here any combination of questing, Sunken Temple, Black Rock Depths will work, but typically around the mid 40s you loose efficiency. At level 55 you can move into Hellfire Ramparts and start gearing your characters up to be played.
Drawbacks include instance lockouts, resets, character proficiency and player proficiency.
Lockouts and Resets. You can only enter 5 instances per 60 minutes, so if Stockades takes 6 minutes to clear you will only be able to power level in it for 30 minutes per hour. These lockouts significantly slow down levels 15-27.
Character Proficiency. When you try to quest with your character legit, you won't have any weapon skill and will be missing vital class specific quests (like Berserker stance). You will also be missing several pieces of vital gear. Imagine showing up to Ramparts to tank with your Aegis of the Scarlet Commander.
Player Proficiency. Ever join a Ramparts group with a Warrior who doesn't know how to tank? If you level a character this way pay careful attention to the mechanics of your class to make sure you don't suck in your first group outing.
Solo Power Leveling
This method maximizes the amount of experience you earn by carefully planning every action before doing it. Players like Joana broke records and won leveling contests with these methods. Having everything planned out can cut the time it takes to level down significantly, preventing you from doing silly things like showing up to Un'Guro Crater without Mithril Casing for the Chasing A-Me 01 quest, or going into Hellfire Peninsula without bringing some Nethergarde Bitter for Gaining Mirren's Trust. Dozens of guides have been written on how to do this effectively and they will help you solo your character to level 70 in the absolute minimum amount of time necessary. The best guide that I have reviewed and used is Team iDemise's, the guide is a collective creation from Team iDemise and comes with instructions on how to setup in-game Addons to make you even more efficient.
Recently Blizzard started a Recruit-A-Friend program which is allowing a lot of new tricks. I wrote a brief summary about it the other day which you should check out if you're going to power level anything. Am I forgetting anything? Share your power leveling secrets with us and let me know what amazing things you've done to level even faster.