It's the resource used to increase the level of abilities like Taunt or Recovery Boost for a Knight. Attack Up has six levels with each level past the first increasing the attack boost by 1% (10-15% at level 1-6). Taunt has 25 levels with each level past the first increasing the damage reduction by 0.5% (4-16% at level 1-25). Speed Up/Down only have one level so +/-20 speed is the most you'll get out of them.
I don't have any of the other abilities unlocked yet (only enough life crystals to unlock two of Recovery Boost, Crit Boost and Guard Boost; not really feeling like spending 7500 sun medals on Weakness Damage Boost right now) but I figure they can be leveled up as well.