A Badass is an elite version of a regular enemy. In addition to having the word "Badass" preceding their names, they can also be distinguished by their glowing eyes, increased size and other artistic deviations like spiked armor. They are not named foes and not to be confused with Bosses.
Strategy
They are tougher than normal enemies and can absorb larger amounts of damage than regular foes. Unlike regular Creatures, non-human badass variants frequently have an extra elemental property associated with them and often have extra attacks. These elementally-powered badasses may drop an Artifact of their elemental type when defeated.
Although they are much harder to kill, the experience they give and the above average loot they can drop generally makes killing a badass more rewarding than killing regular foes.
Playthrough 2
Upon beginning the second playthrough, Badasses will be replaced by the considerably more powerful BadMuthas, and will spawn more often. They are similar to a Badass in most aspects, but compared by level (say a 30 Badass vs. 30 BadMutha) are more powerful. BadMuthas should be avoided until your character is a good few levels above them. You may find them very challenging, and the Bruisers (now 'brutes') a little bit more so at the beginning of the second playthrough.
Playthrough 2.5
Beyond the second playthrough (even though there are no story or missions), the Badmutha enemies become Superbad, and like all the main bosses beyond the second playthrough, they are often level 51. They are a lot tougher than either the Badass or BadMutha enemies, and should not be taken lightly.