If you have access to a small herd of horses, watch the way they interact. They work out their pecking order by pretty tough means. When the dominant horse wants another to get out of his way, he will first threaten by tossing his head and laying his ears back. If the offending horse doesn't MOVE NOW, the next level is for the dominant horse to bare his teeth and threaten a bite or move his rump around and threatening a kick. If that doesn't work, the next thing is either a hard bite or a kick.
Sometimes when a horse bites its handler, it is being a bit pushy and wants a treat - like a spoiled child, and the bite is more of a nip. Sometimes it is trying to assert dominance and the bite is a hard painful bite. It's pretty easy to tell the difference. Either way, it's a display of a lack of respect.
A "spoiled child" bite can be handled by firm but gentle correction, such as firmly forcing him to back away from you by threat and by a firm hand on the halter rope. Consistency in demanding respect by firmness is key.
The dominance bite has to be dealt with like horses do it. I don't mean you have to bite him, like one poster mentioned
, but you have to be aggressive. Forcefully back him until his eyes show he's afraid of you, hit him hard, kick him, or whatever you can, but make him know he's somewhere down the pecking order from you and that you will not tolerate him getting into your space without your permission. Again, consistency is the key. Horses in a herd rearrange the pecking order all the time as horses age and others are introduced. You have to maintain your image as the top-dog. It's not hard to maintain, but sometimes it takes harsh means to get there.
Clinton Anderson and many other trainers get all the dominance and respect issues squared away during ground training. They use whatever force is necessary to get submission and compliance. fixing theses issues is best done during ground training. Once that is squared away, you'll rarely have issues while in the saddle or other activities.