An "alien horse" was a particular type of alien animal, similar to an Earth horse, with an unusual bite.
In 2380, a Starfleet ensign encountered and was bitten by this type of alien horse, causing him to experience accelerated growth on the right half of his body, while simultaneously experiencing reverse aging on his left half. Starfleet considered his condition a medical mystery, calling in Division 14 and the starship Osler to take the ensign to The Farm on Endicronimas V for additional study and treatment. (LD: "Much Ado About Boimler")
This type of animal was only mentioned in dialogue.