Trainers in Skyrim are NPCs that provide training to improve the proficiency of a specific skill. Trainers and their locations are covered on this page.
Trainers come in 3 different expertise levels each corresponding to a different maximum skill level they can train you up to, when you reach or pass that level they can no longer train you. An adept trainer is only able to level your skill up to 50, expert trainers are able to help level your skill up to 75 while a master trainer will be able to level it up to 90.
For each character level you are able to train a total of 5 skill levels. For example training 3 levels in enchanting then 2 levels in smithing would limit you from training any skills until the next character level is reached. The cost of training increases with the level of the skill you are attempting to train in and can get quite costly towards the higher levels.