People are judgment-proof if they lack the resources or insurance to pay a court judgment against them. For example, suppose that a thief steals your car, sells it, and then burns all of his worldly possessions. Even if you sued him and won, you could not recover anything because the thief is judgment-proof.
Plaintiffs faced with judgment-proof defendants rarely given up immediately. Instead, they usually try to find some other defendant who they might be able to sue to recover compensation for their losses. Continuing the above example, if your car was stolen from a restaurant's valet parking lot, you might try to sue the restaurant.
People who are judgment-proof are not necessarily poor. Many states allow people to shield certain assets from judgments, particularly homes. Alternately, a defendant might transfer property to friends or family members so that it is no longer in their name. Other people send their assets outside of the jurisdiction of American courts. However, asset shielding mechanisms do not always work.
See Travelers Indemnity Company v. Bailey for an example of a Supreme Court case involving a judgment-proof wrongdoer. The Supreme Court heard the case earlier this term, but has not yet issued a decision.