As you complete the primary levels of the game (the non-trial levels), you can pull up the menu to find a list of three secondary objectives that must be completed for these achievements. The all have the requirement of clearing all of the pegs, beating a certain ACE Score, and the third objective varies for each level, and can include things such as doing a certain number of a particular skill shot, completing the level without the help of your master (use a different master, or avoid the green pegs), complete the section with a certain master (these appear in the last section of the game), among other objectives.
The objectives can be completed over however many attempts you need to make, however objectives that require you to, say do 5 of the same skill shot, must be performed in a single attempt of the level. Objectives also unlock even if you don't beat the level, allowing you to focus more on a single objective than on the level itself, such as focusing on setting up the level to allow for a huge stylish combo for big points, instead of using the balls available to clear out the orange pegs.
Style Shots (This list can also be found via the how to play or help sections of the in-game menu):
Long Shot: Hit a non-blue peg, travel 1/3 the width of the screen and hit another non-blue peg
Double Long Shot: Get a Long Shot, then travel 1/3 the width of the screen and hit a third non-blue peg
Super Long Shot: Hit a non-blue peg, travel across 2/3 of the screen and hit another non-blue peg
Free Ball Skillz: Bounce the ball off of one peg and into the bucket
Mad Skillz: Have 5 balls in a row fall in the bucket after hitting at least one peg
Crazy Mad Skillz: Have 10 balls in a row fall into the bucket after hitting at least one peg
Extreme Slide: Hit 12 consecutive pegs in a slide
Off the wall: Bounce off the wall, travel 1/5 of the width of the screen and hit a non-blue peg
Kick the Bucket: Bounce the ball off the bucket to hit the final orange peg for Fever.
Lucky Bounce: Bounce the ball off the bucket, then catch it again.
Orange Attack: Hit a large percentage of remaining orange pegs in a shot.
Cool Clear: Hit the final orange peg for Fever, then hit all remaining pegs to clear the level
Double Power Style Shot: Hit both power (green) pegs in the same shot
Wrecking Ball: Get 15 or more pegs with Jeffrey's Bowlder
Spare Pickup: Get the Bowlder into the bucket.
Frostbite: Chain seven pegs with Berg's Deep Freeze
Cool Catch: Catch a peg that is sliding from Deep Freeze, in the bucket
Major Discharge: Trigger 10 Uber Volts discharges in one shot
Charge Caught: Catch an Uber volt shot in the bucket
Ghost Touch: Hit 18 ghostly blue pegs with a Nightshade Shot
Possessed: Catch a Nightshade ball in the bucket.