Broke down or breakdown - a potentially career-ending injury (note this includes but is not limited to life threatening injuries - many racehorses break down and come back sound for less high pressure careers or for breeding).
Broke maiden or break maiden - won a race for the first time.
Across the board - a bet on the same horse to win, place, and show. A way of "hedging your bets".
All out - a horse running as fast as it is physically capable of without regard for tiring out.
Allowances - reductions in the weight a horse carried, for example if an inexperienced jockey is riding or when fillies race against colts (in general, male horses are faster).
Apprentice - an inexperienced jockey. Sometimes called a "bug."
Backside or backstretch - the stable and employee area at a racetrack. Backstretch also refers to the straight part of the track furthest from the stands.
Front stretch - the straight part of the track closest to the stands and the location of the start finish line.
Bearing in or out - deviating from straight, usually unintentionally (i.e., without signal from the jockey or because the horse is misunderstanding a signal).
Distaff - a female horse, especially when discussing breeding. For example, "He goes back to Lady Isabel on the distaff side."
Heavy track - a very wet track.
Juvenile - a two-year-old horse.
Pony - a horse used to lead a racehorse to the gate. (Not all countries use ponies).
Maiden - a horse or jockey who has not won a race.
Morning glory - a horse that performs better in training than in an actual race.
Bute - a common NSAID used as a painkiller in horses.
Rabbit or pacesetter - a horse entered into a race solely to set the early pace and improve the chances of a stablemater.
Rail - the barrier on either side of the track.
Scratch or scratched - remove from a race prior to the start.
Second dam - a horse's maternal grandmother. Grandam is usually used to refer to the dam of the horse's sire.
Claiming race - a race in which any entry can be purchased for a set price. These are the lowest level of officially recognized horse racing.
Sloppy - a track that has standing water on it.
Stall walker - a horse that frets or paces in its stall.
Trifecta - a wager in which the person picks the first three horses in exact order. Very tough to do.
There are lots more!