Dragon Essence (DE) is the material used for promoting your Beast Soul, it can be obtained by killing dungeon bosses in level 50+ nightmare dungeons(DE is not a guaranteed drop, and odds appear to increase on higher level dungeons.), these are bound, and so untradable. DE can also be bought from the Fate Shop.
Beast Soul Evolution
Each stage of the Beast Soul evolves through levels 1-10, when you max level 10, your Beast Soul creature will be promoted to the next stage. Each promotion attempt consumes one Dragon Essence.
Each attempt have a chance of giving bonus progress, the Crit, which gives 5x the current level's normal progress, and the Luck, which gives 10x progress. The Crit and Luck proc rate is shown on the upper right of the Beast Soul window, normal rate is 12% for Crit and 8% for Luck. There you also have a button marked "Boost", which lets you trade 50 crystal for 15 attempts with double the normal probability(Crit rate: 24% and Luck rate: 16%).
You have the choice between doing a normal and a 10x promote, the 10x simply consumes 10 DE and gives you the same amount of base progress as clicking the promote button 10 times. The 10x seems to only do one Crit/Luck roll, which is then applied to the total progress, not one roll for each DE consumed.
Beast Soul Stages
The Stages of the Beast Soul.
Beast Soul Skills
The Beast Soul skills are special passives that are unlocked when you reach certain stages in your Beast Soul development. Currently there are three unlockable skills:
Panzer Shield Lv. 1
This skill is gained when you upgrade the Beast Soul to Lion. It gives a passive increase of 30 to your PDEF.
Vampire Lv. 1
This skill is learned when your Beast Soul reaches the Seiryu level, and returns 5% of the damage you inflict as health. This seems to be calculated from the actual damage in health points done to your enemy, not the damage you hit for, as it seems to be capped(Possibly at 5% the HP of the enemy if you hit for more than it's total health)
Dragon Roars Lv. 1
The Shinryu skill Dragon Roars is a passive with a 5% proc chance to reduce the target's stats by 20% for the 3 seconds.