Okay, I am not an expert at showing by any means, but I've been doing the small local shows for many years.
Showmanship - The handler is being judged at how well they can control their horse. While you don't need to spend an arm and a leg on your outfit (at the world show, then you would) you should be completely neat and tidy. That includes polishing the BOTTOM of your boots (which the judge can see when you trot away from them. Not even one hair should be astray on your, or your horse. You don't need the fanciest silver halter on your horse, but you need one that FITS your horse properly (that's the most important ... that it fits well). You will be expected to walk your horse, trot your horse (possibly both through patterns), ask your horse to pivot on a back leg, and expect your horse to "set up" for the judge like a halter horse. Do all this with a SMILE on your face. You should enjoy showing off your horse to the judge. Make sure to make confident eye contact. Also, know your quadrants. When the judge is in your horse's RF quandrant, you should be in the LF. WHen the judge moves to the RH quadrant, you should be in the RF. And so on. You should not be "dragging" your horse on the leadrope; rather, your cues should be invisible.
I have found this blog to be helpful for Showmanship Success In The Show Pen: Training the Showmanship Horse (Updated)
For horsemanship, the RIDER is being judge on how well they can get their horse to perform manevars. You will be expected to walk/trot/lope through a pattern, including pivots, maybe sidepassing, or even a turn on the fore. And of course, backing up.
I feel that competitive trail relates very well to horsemanship, because they both require a soft, responsive horse, where you can move any part of their body at any time you wish. Trail, you will be asked to open/close some sort of gate. You will have to back through some sort of pattern; typically an L shape. YOu often will trot/lope over ground poles. Here is where it is important that your horse is naturally sure-footed, because you will get docked if they knick the ground poles.
Check out a few videos from the AQHA World Show to get an idea of what you may encounter. But please note that local shows will NOT be this complex .... it's the world show!
Again, any and all the classes, you want to be neat and clean, your horse should be neat and clean, and so should your tack on your horse. You don't have to have the most expensive things, but they need to be NEAT and CLEAN. Presentation is everything.