This is a difficult question to answer, as nowhere in the book is it explicitly stated what exact breed Black Beauty is. We are, however, given some descriptions of his appearance, such as the fact that he is mostly black aside from white marks on his foot and forehead. In general, most readers think of Black Beauty as a thoroughbred.
Aside from the book, we can look at the film adaptations and see what breed of stallion was used to play the titular horse. In the most recent adaptation, released in 1994, Black Beauty was played by Docs Keepin Time, who was an American Quarter Horse. The American Quarter Horse excels at sprinting and is the most popular breed in the United States. While we don't know for sure if Black Beauty in the original story was an American Quarter Horse, this is some of the best information available to us.