AR is used in conjunction with DR to determine the total amount of damage done to the target.  For every point AR is higher than DR (eg: 50 vs 49 is +1, 50 vs 46 is +4, etc), Attack is considered 10% higher.
For players, AR goes up by 1 every 4 levels.  The AR of the weapon used is added to that.  In general, the AR on weapons goes up by 1 every 4 levels.
For monsters, their AR is equal to half their level (rounded down).  Area bosses get a bonus +2 AR.