Chaotic evil is one of the nine alignments in Dungeons and Dragons. The game's 3rd edition Players Handbook calls the chaotic evil character a "destroyer." Indiscriminate violence, cruelty, unpredictable mood swings, and so on are the chaotic evil character's way of life. This is not the sort of person one spends a lot of time with without incident.
To break it down, the chaotic half of the alignment refers to the character's disposition towards law and chaos. Law and tradition mean nothing to her. She cannot be restrained, and she will not tolerate anything that keeps her from pursuing whatever whim or lust pops into her head. She does not play well with others.
The evil half of the alignment refers to the character's attitude towards his fellow man and the common good. It is not that the evil character necessarily does not care about others or does not value life; he just cares about himself and his own life above everything else. He cares absolutely nothing about others except in their capacity to serve him and his irresistable appetites.
Thus the combination of these two elements -- unwillingness to submit to authority plus evil -- produces a psychotic bastard without self-control and conscience. Destruction and mayhem follow her wherever she goes. Chaotic evil characters, in my experience, if role-played as they should be, are difficult to play with because you never know when he is going to get drunk and burn down the inn in which you are currently sleeping (and where all your stuff is stashed) or start murdering people in broad daylight in front of the guardsmen's barracks. Or spit in the face of the necromancer hiring the party to find a powerful artifact to help him become a lich. I loved that one.
Common nicknames for or phrases heard around the chaotic evil character: asshole, "I can't believe you," you son of a bitch, "that's not funny, that's sick," "are you nuts?" "Did I mention I'm not in the room with this guy?"