An animal trainer was someone who taught animals how to follow commands.
In 2365, Jean-Luc Picard was surprised by Deanna Troi's reaction when he offered to find her a gentle horse to ride in the holodeck. She declined his offer, saying she "prefer(red) a mode of transportation that doesn't have a mind of its own." Picard commented that he thought Betazoids would make good animal trainers. Troi explained that it was too easy to become lost in the animal's emotions and thoughts. Picard later noted that horse trainers did not believe in overburdening a young horse with too much weight, a practice he compared to Wesley Crusher's mounting responsibilities aboard USS Enterprise-D. (TNG: "Pen Pals")