"Shaping consists of taking a very small tendency in the right direction and shifting it, one small step at a time, toward an ultimate goal.”

~Karen Pryor~ Don’t Shoot the Dog 

This means that we take something an animal is physically and mentally capable of doing and, using very small steps or increments, teach it to do that specific behavior reliably, and on cue.  Take teaching a horse to rear, for example.  Horses, even foals just days old, already know how to rear on their own.  Our job is to train, i.e. shape, them to do the same thing on cue and only on cue. 

About the three methods mentioned:

Target training is teaching the horse to touch a part of his body to a specific item (target) on cue.  Usually we start by having the nose touch something. Target training is getting to be a familiar concept.  Trainers of whales and dolphins use this technique all the time.  You start by shaping your horse to touch a specific target – it can be your hand, a plastic bag, or a ball on the end of a stick.  As soon as the horse touches the target, you reward.  You start very slowly; perhaps by touching the horse’s nose with the target the first few times.  You gradually progress to moving the target farther and farther away, always keeping it within the horse’s ability to touch it.  You can use target training to teach your horse almost anything.  You can even use it while riding.
 

Free shaping is shaping the horse’s movement without any stimulus from us.  If you want the horse to load into a trailer, you might let the horse loose in a pasture with a trailer in it.  You wait for the horse to look or move toward the trailer.  As soon as he does, reward.  This method works best with clicker training as you can capture the exact moment the horse looks at the trailer.  You continue shaping any movement or attention toward the trailer.  It is very simple, but can take while.
 

Aversive stimulus is using anything that makes the horse even slightly uncomfortable and then removing the stimulus as soon as the horse does the behavior.  It can range from a very slight irritation to extreme pain.  When we are teaching the horse to shake his head “no” by tickling his ear, we are irritating him to get a reaction – aversive stimulus.  We can then shape that behavior to be done on cue with positive reinforcement.  The positive reinforcement allows us to use less and less aversive stimuli to get the horse to do the desired behavior.  This is how most horse training is done.  One can use the lightest touch and still get results.
 

All these methods can be used to shape a horse’s behavior.  You can use target training to teach a horse to load itself in to a trailer You can tap the horse’s leg, which is aversive stimuli, to teach a horse to lift his leg on command, which is the beginning of the Spanish walk.  You can free shape a horse to stand alongside a rail or a mounting block so that you can get on.

We shape a horse’s behavior all the time in training.  We shape the horse’s ability to stop, by teaching them to stop on cue, pulling the reins.  We shape the horse’s ability to go forward by teaching to the move ahead off of our leg cues.  All training, of any animal, is merely shaping the action into a learned behavior.