The Trainer ranks are pretty straight forward. And depending on a trainers rank, they are paid a certain amount each year and gain access to certain benefits on top of that they can already earn through battling others, gym challenges and other competitions.
Novice Trainer - $20,000 Pokedollars
Adept Trainer - $50,000 Pokedollars
Professional Trainer - $80,000 Pokedollars
Expert Trainer - $120,000 Pokedollars
Elite Trainer - $200,000 Pokedollars
Master Trainer - $500,000 Pokedollars
How ranks are attained is pretty straight forward. The ranks are a measure of the trainers overall ability at raising and training pokemon to high levels. The quickest way to raise the ranks is to go through the gym challenge and get as far as possible. The other way, requires time. The trainer needs to have been a trainer for a certain amount of time, and then take a test to measure their overall ability. This is so some kid can't just be given a really strong pokemon by say an Elite-Level trainer and go take the test.
Novice Trainer - Gain A Trainer License
Adept Trainer - Either gain 2 Gym Badges, or be a trainer for a year to take the test.
Professional Trainer - Either Gain 5 Badges or be a trainer for two years to take the test, after having passed the Adept test.
Expert Trainer - Either gain 8 badges, or be a trainer for four years to take the test and having passed the Professional test.
Elite Trainer - Reach at least the top 8 in a Pokemon League, or be a trainer for six years to take the test, and having passed the Expert Trainer test.
Master Trainer - Win a pokemon league competition and then hold your own against an Elite Four member, this is to make sure it isn't just luck through a weak competition. Or, be a trainer for ten years and passed all tests from Adept to Elite to gain the right to take the test for this rank, which again, among many other things, concludes with you having to battle and hold your own against an Elite Four Member.