You often see this in the movies (where the horse has been trained to do it on cue). Why does a horse paw the ground?
There are several reasons:
1. The horse wants to roll. Horses that are going to roll will often test the ground they intend to roll on by pawing at it first.
2. Begging or food seeking behavior. Begging in domestic horses often consists of holding a foreleg up partly folded, but some horses may paw to beg.
3. The horse is overly excited or frustrated - you might see racehorses paw at the start line, some show horses paw when it's close to their turn to go in. A horse that wants to run and is prevented from doing so by his rider may paw. Alternatively, nervous horses may paw or stomp the ground.
4. The horse is bored - this is why horses in lessons might paw when waiting a long time for their turn, or a stalled horse may paw for want of anything better to do.
Pawing at water is a slightly different thing. Some horses just love to play in water and I've witnessed horses paw at water more on a hot day, so they may be trying to cool off. However, some horses will also roll in water, so be careful.