The melee is very minimal. Mordecai's melee attack is a machete, I believe, but the animation happens so fast that you can barely see it; kind of disappointing actually. I dunno about the other characters.
There are skills and weapons and stuff that increases your melee damage though. for instance, Mordecai has a skill that increases melee damage and gives every melee strike a chance to be a "fatal hit" or something, doing assloads of damage. Then there are the weapons with blades on them; pretty self-explanatory.
But really, this game is about shooting guns. Melee is sort of an emergency only tactic unless you use one of aformentioned skills/weapons that make it actually worthwhile (or are playing as Brick, obviously). Also, melee is good for one-shotting super low level mobs that you don't want to waste bullets on.
