A chaotic evil character does whatever his greed, hatred, and lust for destruction drive him to do. He is hot-tempered, vicious, arbitrarily violent, and unpredictable. If he is simply out for whatever he can get, he is ruthless and brutal. If he is committed to the spread of evil and chaos, he is even worse.
These characters are the bane of all that is good and organized. Chaotic evil characters are motivated by the desire for personal gain and pleasure. They see absolutely nothing wrong with taking whatever they want by whatever means possible. Laws and governments are the tools of weaklings unable to fend for themselves. The strong have the right to take what they want, and the weak are there to be exploited.
Thankfully, their plans are haphazard, and any groups they join or form are poorly organized. Typically, chaotic evil people can be made to work together only by force, and their leader lasts only as long as he can thwart attempts to topple or assassinate him.
As an example a demented sorcerer pursuing mad schemes of vengeance and havoc is chaotic evil. Chaotic evil is sometimes called "demonic", because demons are the epitome of chaotic evil.
Chaotic evil is the alignment which represents the destruction not only of beauty and life but also of the order on which beauty and life depend. It also the most unstable and destructive form of chaos or evil (compare and contrast with lawful good), due to losing ones self to impulse and desire.
The Infinite Layers of the Abyss (of the Lower Planes) embodies this concept.