What I know so far from dev posts during beta or just generic guessing-work:
Power Rating: This is supposed to denote how "powerful" your item is. Generally speaking, a level 3 item would have a slightly higher power rating than a level 1 item. Because of this, the level 3 item should have better stats, such as higher defense or attack, etc. However, depending on personal preference (maybe you want to use a lesser defense item because it also has +mana), a lower power rating item could very well be more desirable. Also, at this point, their itemization is still not perfected, and several jobs have lower level gear with better stats than the higher level versions.
Star Boost: I believe this one only occurs right after you receive a level. For a short time afterward (10-15 mins I think), you have a +5 to this, maybe a 5% bonus? As far as I know, it's a slight increase in star gain.
Coin Boost: Obviously something that gives you more coins from money piles. I've never had this before.
Defense/AC: This is the blue shield icon. From the classic MMO view, this should help protect you some against how hard a creature can hit you for, mitigating that damage some.
Energy: This is the blue icon with lightning jags (normally known as mana). It increases your energy by X amount (whatever is listed).
Health: This is the heart icon. It increases your health by X amount (whatever is listed).
Attack: This is the sword icon. It adds to your attack rating (or is your attack rating if you only have it from a weapon). I'm fairly certain the amount of damage you do when you strike a creature with a normal hit is the same as your listed attack in the "About Me" section of a combat job.
Matcher's Timing: This is the watch icon. I'm not 100% sure what it does to be honest. My two guesses are that it adds more to your time when you make matches, or that it makes your timer fall more slowly.
Matcher's Skill: This is the blue star icon. I assume it gives you more stars during the minigames that grant stars, but again, not certain.
Matcher's Advantage: This is the hammer icon. I have no idea what it does honestly.