Trainees are a class type exclusive to Fire Emblem Gaiden/Echoes: Shadows of Valentia and Fire Emblem: The Sacred Stones that has multiple promotion stages. The classes start out fairly weak, but if you allow them to class change into first and second tier classes, they will prove quite useful.
In Gaiden, the only Trainee class in the game is the Villager. The class is notable because of its ability to change into almost every other class in the game (male Villagers can promote to Mercenaries, Cavaliers, Archers or Mages, while female Villagers can promote to Cavaliers, Pegasus Knights, Mages or Clerics). In addition, Dread Fighters can promote back into Villagers.
In The Sacred Stones, the classes are Pupil, Journeyman, and Recruit class. All together they can have up to fifty levels. If both routes in game has been completed, the classes gain the ability to class change into a Super Trainees. They have low Con, meaning that they are slowed by heavier weapons, but compensate for it with other abilites.
There was also the 'Probation' class, which was a class that existed in early development of The Sacred Stones and was probably a Trainee type class.
Story-wise, the reason for these weaker classes are generally attributed to young age or inexperience. According to the Support conversations from The Sacred Stones Amelia, Ewan, and Ross are all the same age and are training to become a real Knight, Sage, and Warrior respectively.