I know that the Orlov Trotter is quite a rare breed of horse, but I have not heard of many others. Is there a large list or only a select few and why are they so rare?

There are many breeds of rare horses. Many of them are rare as they have been developed in mainly isolated areas from select stock and not sold to outsiders or bred elsewhere. After horses have attained a recognizable difference from the foundation stock that can be reproduced it can be considered a new breed (to a degree, there are a couple more rules I'm sure). There are also breeds which are rare as the result of disease, resource limitation, and other limiting factors.

Basically, if the breed no longer has a use, then it becomes rarer. Many of the draft breeds are in decline for that reason. These days people have tractors to do their work instead, though there are still a few enthusiasts around.