Pantomime horses are a stock British comedy character played by two people in a horse costume. One person plays the front legs and head, the other bends over and plays the hindquarters and hind legs. They come from the tradition of comic folk and street theater going back to the Middle Ages. The comical plays are staged in every town and city, particularly at Christmas, by professionals or amateurs. The exact setting and story varies (and may be loosely based on a well known story like Cinderella or Peter Pan), but always features a boy loves girl/good versus evil storyline and well-known stock characters. An animal character - often a horse or cow - is typical, and plays a sympathetic & helpful role. This kind of pantomime has no relation to "mime" (the silent acting of Marcel Marceau, etc.), and is little known outside of Britain and Australia.
As this site explains, the appeal of these performances is hard to explain to foreigners:
The plot is very simple: The girl dressed as a boy who is the son of a man dressed as a woman, will win the other girl (surprisingly dressed as a girl), with the assistance of a person(s) dressed in an animal skin. Try explaining that to an American, German or someone from the Far East.
These plays are meant to be fun and appealing to all ages. Slapstick and audience participation are always a part of the performance and many celebrities and well-know actors join in the fun.
This clip gives you a sense of the performances, though it doesn't include any horses:
Good video clips of pantomime horses are hard to find! This one will have to do for now: