Awakenings are used to make your equipment more powerful.  Awakenings stats can only be obtained on certain items as well.  You can get a max of 3 awakenings on each part.  But be careful, some awakenings will be negative and effect your character in a negative way.  A list of awakenings:

Weapons/Shields/Suits can get any awakening effect.

Helmets can get STR/STA/DEX/INT/MP/FP/HP

Gauntlets can get Attack/Critical Rate/Attack Speed/Decreased Casting Time/MP/FP

Boots can get Increased Critical Damage/Speed/Defense/HP/FP/MP

Awakening Chart:

Awakening Amount you can obtain
STR/STA/DEX/INT 1/4/710/13/16/19/22/25/28
CRIT/DCT/SPEED 1/2/3/4/5/6/7/8/9/10
ICD/A SPEED 1/3/5/7/9/11/13/15/17/19
ATTACK 5/20/35/50/65/80/95/110/125/140
DEFENSE 4/16/28/40/52/64/76/88/100/112
HP/MP/FP 50/100/150/200/250/300/350/400/450/500