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The Mastery system is a new game mechanic added to the talent system in World of Warcraft: Cataclysm. It has undergone extensive revision since its announcement at Blizzcon 2009. With the dramatic revamp of the talent system in preparation for Mists of Pandaria, mastery now represents a general bonus from items that gets applied to specialization (chosen at level 10) as single "Mastery: " ability which determines how the bonus is converted to some benefit.
The mechanic hasn't changed dramatically, but theoretically, the system is much simpler since the benefit from mastery is clearly stated in the single specific ability gain from choosing a specialization.
As of Cataclysm
This section concerns content exclusive to Cataclysm.
Your mastery is a passive benefit to your character based on what talent specialization you chose. As noted in the tooltip for each paladin mastery here on Wowhead, you can increase the effectiveness of your mastery bonus by stacking items with mastery rating: http://www.wowhead.com/talent#s
Some classes or talent specializations desire mastery rating much more than others, depending on how powerful the bonus is to their role, but the basic idea is that this passive skill is attractive to you because of the talent specialization you chose.
This is used in items with a level requirement of 78 or above[1] or item level 272 and up.[2] For example some uncommon items will have mastery by default, just like stamina or intellect.
Mastery is in addition to talent trees, this helps the Mastery buff gained at level 80.
How it works
Basically, the Mastery rating means nothing unless you have the appropriate Specialization which is a new concept introduced in Cataclysm. Specialization is basically a form of locking in your talent tree and doesn't happen until level 80.
As of Patch 5.0.4, the points based system was removed. Now, each specialization has a specific ability that determines how the mastery bonus benefits the character.
At level 78 and above, items will start dropping with Mastery as a stat.
Mastery bonuses get applied based on specialization ability gained that starts "Mastery:" in the name.
Mechanic in Cataclysm
This section concerns content exclusive to Cataclysm.
Here was a summary from WoW Insider of the Mastery system circa Cataclysm:[3]
You choose one tree at level 10 and then can only add talents to that tree until you have put 31 points in it. Then the other trees open up.
At level 78 and above, items will start dropping with Mastery as a stat. Once Mastery is learned from your class trainer, the stat will give bonuses based on the tree specialized in.
The Mastery bonuses will be flat percentages and will no longer be based on the number of talents spent in a tree.
Talents will alternate with skills when leveling. So you will get a talent point "about" every other level.
So, you need to keep a close eye on the bonus you get in your Mastery talent and decide if the conversion of Mastery rating to the bonus is worth it to you. Wowhead has a nice list of Mastery talents.