A monk trainer is an NPC that offers monks the opportunity to reset their talents and change their specialization. A general rule-of-thumb for finding the local Monk trainer is to search for either a peaceful, secluded area where one may be meditating, or in a local tavern putting their drinking skills to the test. Because the leaders of the respective Alliance and Horde Pandaren factions are Monk trainers themselves, keep an eye out for the turtle mounts and Pandaren architecture!
The monk trainers are unique among class trainers in that they are all pandaren (with the exception of the troll Zabrax, who has apparently been trained by the nearby Tsu the Wanderer). In the case of the starting area trainers, this usually makes them the only class trainers which are not of that area's race. They are also all level 90, despite the starting area trainers being found in areas where most other NPCs are somewhere around level 5.
Following the two pandaren groups' joining of the Alliance and the Horde, monk trainers traveled to each allied race's starting area to teach the ways of the monk, and also to learn the ways of the other races. These pandaren are distinctive characters, and in many cases have clearly chosen (or been chosen) to train that specific race due to their personalities. Some examples:
Gato is a well-armored pandaren sent to train the orcs.
Ting, Strong of Stomach is obviously well-chosen to handle life among the Forsaken. Her strong stomach impresses even Archibald Kava, when he puts it to the test with a drink of dubious safety.