Awakening is pretty much a 2nd job advance... except u can only choose from one job which depends on what sub-class you are. Asura's awaken into Dark Knights, Summoners awaken into Gaia's, Soul Benders awaken into Soul Reapers, etc.
When you awaken, you unlock your classes last 2 skills (1 passive skill and one active skill) These skills pretty much make you look awesome in every way possible.
You can begin your awakening quest when you reach lvl 48. From there go to your class instructer and follow the quest lines untill you awaken.
All classes will require 500 VP (Victory Points) which can be obtained from PvP or Tower of Illusion/Tower of Death in order to awaken.
The awakening quest line for asura's is the following:
1.Talk to GSD.
2.Bring GSD 10 White Cubes, 10 Red Cubes, 100 Superior Irons, and 100 Superior Hardeners.
3.Collect 40 pieces of Terranite
4.Bring GSD 500 victory points.
5.Clear King's Relic 3 times on any difficulty.
6.Clear Vilmark - Area 50 4 times on any difficulty.
7.Talk to GSD
If your wondering what Terranite is, its a material that only drops from the Vilmark Dungeon.
King's Relic and Vilmark both require Spirit Crystal to enter. You will get plenty of Spirit crystals on your way to level 48 so dont worrie about thos.