Forging is when the front of the hind foot hits the bottom of the front foot, usually at the trot. It's considered a fault (and can sometimes result in injury to the horse). It's most commonly seen in young horses that are going through an awkward phase - and they often grow out of it. In some cases, it's a permanent conformation fault, most often because the horse has a short back and long legs. Out of shape horses sometimes also forge because their muscles are stiff.
The obvious symptom is a clicking sound, especially if the horse is shod.