The Coach is an NPC and almost always a man, though rarely the coach will be a woman. The coach will yell at a Sim to stop whatever he or she is doing and work out, but that Sim may not do so right away, in which case you can cancel the Work Out action before he or she starts working out. While the Coach can be very irritating, a locked door can keep him out, and if you can have his target ignore or avoid him long enough, he will give up and leave.
Supposedly, the Coach targets unfit Sims. In practice, he often targets very fit ones.
Coach is also a job in the Athletic career track. This job has no relationship to the University NPC.