Bull nose: front of hoof bulges out, goes with steep pasterns.
Dish foot: concave hoof wall even when the toe is the correct length, a sign of the horse having had founder or a high fever for a long time.
Underslung heels: the heel angle is lower than the toe angle, the horn tubules are not all parallel. Goes with weak heels or something causing the horse to stand on it's toe for a long time, like sore heels.
Broken back: hoof angle is less than the pastern angle. Increases stress on flexor tendons and the deep flexor tendon which goes over the navicular bone (perhaps one of the contributing factors of navicular). The toe may be too long, horses wearing shoes have toes that tend to grow faster.
Broken forward: The toe angle is greater than the pastern angle, such a horse may have steep pasterns.
Club foot: abnormally steep hoof angle (>60 degrees). If it's genetic it cannot be changed, but if it it because of contracted tendons on an injury to the heel it is possible that it can be fixed.
Flat foot: sole is flat, not concave. The sole is easily bruised because it is always bearing weight.
Dropped sole: even worse than a flat foot, it is impossible for such a horse to be sound because it is always walking on it's heels.
Cleft: a horizontal crack in the hoof which grows out.
False quarter: a vertical line on the hoof that is always there, the result of an injury to the coronary band, it is permanent.
Contracted heels: heels are too close. When the toe is measured across and one inch back and the heels are measured across a quarter inch forward, the two measurements should be equal. If the heel width is less than the toe width the the horse has contracted heels. Caused by not enough movement (the heels need to be able to expand and contract during movement, when the horse doesn't move around much for several weeks the hoof is allowed to get used to a permentent state of contraction) or a trauma to the leg (if the horse is put on stall rest and doesn't get to move around enough). Other causes are things which hinder the spreading of the heels - too small shoes, not enough moisture, and nails placed too close to the heels. This condition does not happen over night, it takes a couple weeks to develope.