Some of these terms are heard in the show warmup ring, and in most dressage riding lessons.
More from behind
Since the stride starts from the back legs, and good dressage is energetic, when you hear More From Behind, you need more energy. Use More Leg to get those hind legs underneath you.
More Leg
If you want More From Behind, you will need to use More Leg, hopefully for just an instant. Your lower leg is the cue to the hind leg of the horse, because it is the closest part of your body. Press the gas pedal.
Get Him In Front of The Leg
See More From Behind. The horse is not listening to your leg.
More Active
See More From Behind.
Over The Back
Your horse is showing some stiffness, perhaps in his back or in the base of the neck. You need a rounder profile, a rounder shape Over The Back. Think of the silhouette of a muffin versus a pancake.
Deeper
See Over The Back. Lower the neck of your horse to relax and stretch the muscles at the base of the neck.
Get Him More Through
See Over The Back. Try riding Deeper.
Rounder
See Over The Back. Try riding Deeper.
Supple Him
See Over The Back. Try riding Deeper.
On The Bit
An energetic, round outline to your horse that feels responsive, comfortable and obedient to ride.
On The Aids
See On The Bit, your trainer does not want you to focus just on the headset of the horse.
More Collected
If your trainer is saying this to mean On The Bit, you need a new trainer.
If this is not the case, your horse is already On The Bit and needs to shorten his outline from nose to tail without losing energy, and sit back on his hind legs a bit more. He will take More Active steps, like a coiled spring.
Half Halt
This one's harder to define because there is variation between riders, horses and situations. You need to call your horse to attention with a brief, subtle signal. If he is already Through, he will Collect himself more. A Half Halt will keep him On The Bit. The rider might briefly squeeze one or both reins, might sit taller, or heavier for a moment, might squeeze the upper thigh, or these in combination. A Half Halt always has a corresponding release. You Half Halt to prevent losing his attention, to prepare for a movement, or to improve his balance. If your horse responds, he's On The Aids. It's the most subtle of transitions.
Looking this over, I'm thinking a Dressage Bingo game might have some fun potential for the next time you're watching a lesson!