What does it mean when someone says that they are someone's cousin, "twice removed"?

I’ve heard that expression many times, and it was used by the child-bride of singer Jerry Lee Lewis, when she tried to claim that she wasn’t as “biologically close” to him, as was being reported by the Press. (She was his cousin.)

Is the “removal” process something that naturally occurs in families, or do the courts play a role in it?