1. A term used by intolerant people or by the elite, who find some need, probably from a lack of self esteem, to marginalize someone whose opinion differs from their own. Usually they would define a mouth breather as someone who is to stupid to "get it". A a socially backward individual.
2. Someone who suffers from allergies or other condition that prevents breathing through the nose.
These days, someone who supports President Bush or the NRA, would be described by liberals as a mouth-breather.