The Battle Aura is a Special Skill which requires 1 SP (Stamina) block to be used. It will enhance you with nice bonuses, like +atk, +def, +atk rate, +def rate.
The Battle Auras available:
1. Earth Aura
2. Wind Aura
3. Water Aura
4. Ice Aura
5. Fire Aura
6. Thunder Aura
There are around 6 Auras. They have different looks, but remember, they give the same stats/bonuses.
So, after you have the stats for lvl 20, go to your instructor to take the quest. If you don't have the stats, you have to rank up your rank. If i recall correctly back when i had lv 20 i was lacking 1 dexterity. I ranked up my magic, which gave me some bonus stats, so i had the stats required for the quest.
There is an image which shows you what to click/say to the NPC, and as result which Aura you will get.
I take no credit for the picture, because i didn't make it. I only made this little guide, so have fun... And choose your aura carefully to fit your needs!
EDIT: A picture with the Auras:
http://img214.imagevenue.com/img.php..._122_150lo.jpg