Game Mechanics Guide
Most monster have two types of skills - an active skill and a leader skill. A monster's active skill is an effect that is available to activate after a certain number of turns have passed (the active skill's 'cooldown' time). You can reduce the cooldown time of a skill by leveling up the skill, thereby making it available sooner. The methods for this are discussed more thoroughly here. Active skills come in many different varieties; they can boost the damage of certain attributes or types for a number of turns, change certain orbs to a different color, change all the orbs on the board, etc. Almost all monster have an active skill, except for the very first ones you come across in the beginning dungeons. Also, a monster's skill does not necessarily remain the same as it progresses through all of it's evolutions - some later evolutions of monsters will have skills that are different than their base forms. For example, Commerce Deity Hermes has the active skill called "Double Attack Stance-Wt" which changes wood and heart orbs to water orbs. However, Commerce Deity Hermes can be evolved into Awoken Hermes, and the two monsters have different skills. If you evolve your Commerce Deity Hermes into Awoken Hermes, he will have the new active skill "Dream Water", which changes wood and heart orbs into water orbs and also increases skyfall chance of water orbs for 2 turns. Always make sure to check if your monster's skill will change before you evolve it. This is especially important to note because, if a monster's current skill changes to a new skill, it will lose all of the skillups it gained for the first skill, and the new skill will start at level 1. So, continuing on with out Hermes example, let's say that your Commerce Deity Hermes was max skilled, meaning that "Double Attack Stance-Wt" would be ready after 8 turns. If you evolve him into Awoken Hermes, he will have the new active skill "Dream Water", and it will be set at level 1, meaning it will be ready in 12 turns. Basically, if the name of a monster's active skill changes, it counts as a new skill, and the existing skillups won't carry over to the new skill.
Now, a few details on active skills that enhance your damage. One important thing to note is that you cannot enhance the same type or attribute twice on the same turn. Here's an example - let's say you use Loki, the Finisher's active skill, which gives dark attribute monsters 1.5x attack for 3 turns. Then, on the same turn, you use Founder of Nations, Okuninushi's active skill, which gives 1.5x attack to dark attribute monster for 1 turn (as well as delaying enemies for 1 turn). These two damage boosts will NOT stack - the boost from the second skill will overwrite the first, and you will only get 1.5x dark attribute damage for 1 turn. Both Loki's and Okuninushi's skills enhance dark attribute damage, and you cannot "double dip" with these skills; only one can be active at a time. Now, let's change the example a bit - let's say you use Loki, the Finisher's active skill, and then use Moonbeam Fang Witch, Lilith's active skill, which includes a 1.5x attack boost to Devil-type cards for 1 turn. In this case, these two skill can be active at the same time - one will not overwrite the other. This is because Loki's skill boosts attack for an attribute (dark attribute damage), while Lilith's skill boosts attack for a certain type (Devil-type damage). Since these skills aren't boosting the same type or attribute, they can exist simultaneously, and any monster that is both Devil and dark type will get a 1.5*1.5=2.25x attack boost from both skills.
And finally, one last thing to know about active skills: you can't use a skill when it wouldn't do anything. Let's say you currently have a board with no heart orbs on it, and you have a Samurai Goblin on your team with a fully charged skill; the game won't let you use the goblin's skill, since it would have no effect. Samurai Goblin's active skill changes hearts to fire orbs, and in this example we assumed a board with no hearts on it. There are no hearts to change, so the skill would produce no effects, and thus the game won't let you use it.
The most important thing to know about damage in PAD is that, except for an enemy's Armor stat, everything is multiplicative. This means that even a small additional multiplier will go a long way! Trying to manually figure out the final attack value of your team, or even a single monster, can be difficult since there are so many factors that affect your damage; enhanced orbs, total combo count, awakenings, elemental resistances, and many other mechanics come into play. If you want of figure out how much damage a monster is going to do, the best method is to go into the Endless Corridors dungeon (the bottom one under the 'Special' dungeons tab) and test things out. The Endless Corridors dungeon doesn't require any stamina, so you can go in, make a couple combos, review your damage, and then exit from the menu. There's also a bunch of math you can do to figure out what your damage will be manually, but that's a whole other discussion.
Leader skills don't "Double Dip". For example, a busty Hermes boosts Water and Light monsters, but won't give two boosts to a Water Dominion Gabriel, which is both Water and Light. However, since Gryps Rider is Dark and Light, and Hermes boosts Light, Gryps Rider's dark attack will still receive the boost. Leader skills stack multiplicatively - if you are running Hermes as you leader and another friend with Hermes, the resulting attack boost will be 2*2=4x attack to all water and light monsters (and 2*2=4x HP to all water and light monsters, as well).
A Pre-emptive attack is an attack launched by an enemy at the beginning of a turn, before the enemy's "official" first turn. These become commonplace as you start taking on harder dungeons. Pre-emptive attacks come at the very start of the turn, before you have a chance to use any skill or launch an attack. You can reduce or eliminate the damage from a pre-emptive attack by using a defensive skill on the turn before you get to a floor with the attack.
WORK IN PROGRESS, SORRY!