For Crown Law guide, go here: Crown Law (guide)
Crown Authority decides how much power the king has, and affects several aspects of your realm.
The first, and one of the most important, aspect crown authority decides is the least amount of levies your vassals are obligated to provide. This starts at 0% at Autonomous Vassals, and increases by 20% per level, to a max of 80% at Absolute Crown Authority. This overrides both Opinion and Levy Law as long as it is higher. Higher crown authority thus means that you're able to call on larger levies in times where your vassals dislike you, for example right after succession.
Crown authority is also needed to set some of the more desirable succession laws. For example, you need Medium authority for Seniority, and High for Primogeniture. As a point of interest, you can lower the crown authority afterwards without losing the succession law. Crown authority can be increased by a ruler so long as his realm vote it through, this can be done multiple times supposing a ruler deem it worth the long wait in between.
Further, each step of crown authority gives the king privileges.
Limited Crown Authority
At Limited Crown Authority the king gets the ability to revoke titles and the ability to appoint generals. This level of crown authority is extremely important, as this will let you strip a single title from any vassal that rebels, thus allowing you to redistribute power in a way that makes the realm more stable. Second, you will no longer risk being stuck with horrible commanders in an army stack, as you'll be able to appoint someone more suited to the role. At times, this can win you fights, and potentially wars.
Medium Crown Authority
At Medium Crown Authority the crown gets another two important privileges; free revocation of titles from infidels, and no fighting between vassals. By being able to revoke titles from infidels and heretics you'll no longer have to deal with vassals having large Infidel/Heretic penalties, and you'll be able to redistribute their titles as you see fit. No infighting between vassals means that vassals will no longer be able to go to war against each other except when rebelling against their liege, thus you'll almost always have their levies available when you need them. It will also prevent vassals from getting powerful via intra-realm warfare.
High Crown Authority
Upon passing High Crown Authority titles can no longer be inherited by anyone outside the realm, so you will no longer lose parts of your realm due to inheritance. Foreign nobles will no longer be able to snag your lands from right under your nose, and inheritance gets more predictable.
Absolute Crown Authority
Absolute Crown Authority gives one final privilege: vassals can no longer go to war (except to rebel) at all. This means that your vassals will no longer be able to grow powerful by taking land from outside the realm, and makes them thus more manageable. The drawback is of course that you now cannot grow that way either; you will personally have to start all conquests.
Negative Effects of Crown Authority
Increasing crown authority will not only give privileges, it also brings some severe penalties. First and foremost are the opinion penalties. At Autonomous Vassals you'll get +5 opinion with all your noble vassals, but this rapidly goes down. At Limited, you get -5 opinion; at Medium you get -10; at High you get -20; at Absolute you get -30. This makes vassals less willing to provide tax and levies, and more likely to rebel.
Upping crown authority also increases risk of rebellion directly. At Autonomous Vassals all vassals will have a -20% chance of rebelling, but this increases by 10% each level, ending at +20% at Absolute Crown Authority. Beyond this and the opinion penalty however, there are very few drawbacks. In total you get -35 opinion from going to max authority, and +40% rebellion risk (+17.5 to 35 from the opinion hit, for a total of 57.5 to 75%).
Due to how severe the rebellion and opinion penalties are, I would not recommend going beyond Medium Crown Authority. At this point you've got the most important privileges without incurring too large penalties. Unless you often lose land to inheritance there is no real point to going to High Crown Authority as it is seldom worth the opinion and rebellion hit. One should note that every vassal will get -30 opinion of you for five years if you increase crown authority.
Contributed by Meneth and edited by Blitz.