So, you've all seen those scenes where the hero gets off to check tracks, or whatever, and leaves his horse just standing there.
The question I got was "will a horse actually do that?"
The answer is: Yes, if trained to do it. Modern riders call it "ground tying" - training a horse to stay where it is put without any physical restraint.
A horse that has been properly trained to ground tie will wait calmly while its rider or driver (this training is particularly important for harness horses, who need to stand while the wagon is loaded and unloaded, etc) does whatever they have to do and not move until given permission to do so. It is taught, simply, by putting the horse in a location and moving them back there if they shift position, gradually increasing the length of time and how far the handler moves away. This is such easy training I did it when I was 14.
So, the answer is, a well trained horse will wait for his rider to pick up whatever he dropped, open and close a gate, set up equipment in an arena or even eat lunch. A well trained horse will also stop and stand if his rider is unseated, unless given some reason not to.
What if the horse isn't well trained? Most likely the horse will move off a short distance and start grazing. An unattended horse will generally not run unless spooked or chased. They'll head for the nearest bit of grass and chow down.