There are a total of 16 courses on the disc to master.
In order to master a course you must do the following;
Score a -6 (6 under par) round
Have a hazard free round (no bunkers, water, out of bounds, etc)
A total of 60+ FIR (Hit the fairway from your tee shot)
A total of 80+ GIR
A total of 20+ One Putts
A total of 100+ Pars or better
A total of 30+ birdies or better
A total of 5+ Eagles or better
All of the requirements that say total can be done over a number of rounds, just make sure you save your progress before turning off your Xbox.
This will take an absolute minimum of 6 full 18 hole rounds because of the 100 par or better requirement.
You can view your progress on each course, by going to the Main Menu --> Game Modes --> Course Mastery.
I suggest playing the Road to the PGA Tour as you can make good progress to Course Mastery achievements by doing this anyway, plus getting other achievements on the way.