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I worked in high school with rescue horses, some VERY mouthy ones among them. There were two main issues: some had been hand fed treats and would nip in expectation of a tasty snack. The others were, as your question indicates, biting for dominance or because of other behavioral issues (e.g. when being saddled or having their feet handled). 

For snacks, the first thing to do is stop hand feeding. Always drop treats into a bucket before presenting them.

In both cases, however, you need to clearly communicate that the biting is absolutely off limits. In general you will need to physically assert your own dominance, "bite back" so to speak. Depending on the temperament of the horse this may be more or less difficult; with a quiet, non-confrontational horse a few iterations of blocking the bite attempt with an elbow plus a firm "no" can be enough. Normally I would use the rubber end of a carrot stick to block any bite attempts - a head swinging around would lead to a "thonk" against the rubber butt instead of a chunk out of my arm. 

Note that this is not punishment - it needs to interrupt the behavior, not happen afterward as a response. Pay attention to the ears and other body language to predict when a bite attempt is imminent. Always combine this with a verbal "no." If you can show the horse that biting attempts consistently lead to an unpleasant result, it will begin doing it less. 

If a specific activity (e.g. saddling) seems to turn a normally relaxed horse into a shark, it may be due to prior mistreatment or stress rather than a dominance play. In this case slowly retraining and desensitizing the horse from the ground up to address the root issue may be preferable to just focusing on the symptom.