If it is offered, it will usually be just one of the classes in a normal hunter division.
What can you expect from a handy hunter course? It will vary from your usual inside-outside-diagonal-outside-diagonal hunter course in that it will require the horse to be more "handy". A handy horse is one that is very easy to maneuver around a course. The judge will want to see a horse that can handle tighter turns more easily and take some chances while being very responsive.
Different elements that might be seen in a handy hunter course are gallop jumps, rollbacks, trot jumps, a gate to open and/or close from the horse's back, etc. You might even be asked to dismount and lead the horse over a jump. The judge might also want to see you take chances by riding a rollback turn that isn't specifically asked for on the course diagram, the same as you might see in an equitation class where there is a clear option between going inside a jump to make the turn or going all the way around for what would be a more typical hunter turn. Additional risky elements that should be rewarded if done well are heading straight to the first jump without circling, and easily coming down to a walk before the in-gate at the end of the round without circling.
Here is the official EC rule concerning handy hunter classes:
ARTICLE G315 HANDY HUNTER
May be offered as one class per division. Course should vary from the normal hunter class routine to include elements that show rideability and handiness. e.g. turn back, trot fences, option lines. Not recommended for green horses or novice rider classes.