Registered:  Each individual breed of horse and pony has a registry, or a list of its members.  The registered horses can then trace their ancestry.

Remount:  Young colt just broken in.  (Not sure if this is what is meant in the places I have heard mention of "army remounts" and "cavalry remounts," but would have to go back and look at those sources!)

Ride straight:  In hunting, to jump obstacles as they come.

Ridgel:  Or ridgeling; this term refers to a horse that is half-castrated or with only one descended testicle.

Riot:  In hunting, hounds are said to riot when they run after game other than the fox.

Roach back:  Arched back (also known as hog back).

Roman nose:  Head with a convex nose line.