The title means what kind of trainer they are
Take Ace Trainer, for example.
They are Trainers who frequently travel around the world to meet new Trainers and take part in Pokémon League competitions.
They generally use strong and evolved Pokémon, have a diversified and balanced party, and their artificial intelligence is generally very good compared to other Trainer classes in their games. Like Gym Leaders, they often have healing items, and they also use type advantages to their advantage. They are usually seen during the mid to last part of the game in Victory Road, as they are presumably on their way to challenge the Elite Four. In the Battle Subway, some of them employ a set strategy revolving around the Ability Intimidate. In Generation VI's Battle Maison, they have unspecialized teams that may occasionally include Latias or Latios.
These are kind of the "elite" trainers, one that you'd encounter at the Victory Road or such.
Others, like Bug Catcher, mean that they catch and train bugs.
A full list of the different kinds of trainers can be found here
Hope it helped!