Quarter Horses are the most popular for running barrels, as they can turn quickly (being bred for cattle work) and are the fastest breed in the world, although they lack endurance.
Some Thoroughbred crosses (Appendix) do well, but as a whole TBs simply cannot turn as well.
Arabians just aren't built for it. Too small and skinny, and they can't suck back on their butts. I have seen some pretty decent Arab barrel horses, but never one that wins.
Quarter Horse style Paints also do well, but they aren't as popular. Same with Appaloosas.