Pleading the Fifth
The Fifth Amendment to the United States Constitution provides you with the right to say nothing that might incriminate you or in any way link you to a crime that's been committed. You may refuse to answer any questions that are put to you in this regard, and you are always free to refrain from volunteering information. This is also sometimes referred to as the constitutional right against self-incrimination. When someone "pleads the Fifth," either in criminal procedure or in jest, it means that they are invoking the right not to condemn themselves with their own words (or make themselves vulnerable to condemnation).
Perhaps the most powerful application of the Fifth is in criminal trial, where a criminal defendant always has the right to refuse to testify before the court. Neither judge, nor jury, nor prosecution, nor even the defendant's own council can force the defendant to take the stand in his own trial, if he is firmly resolved not to. If the defendant does choose to plead the Fifth, the jury is forbidden from considering it in their deliberations. However, if the defendant does choose to take the stand in his trial, then it is generally held that he has waived his right against self-incrimination: he will be required to answer the questions put to him like any other witness.
Alternately, witnesses may plead the Fifth by refusing to answer certain questions while they're testifying in someone else's trial, if they believe the answers might incriminate them. This is because, unlike a defendant, witnesses can be forced to testify in court with a subpoena - but this shouldn't deny them their right to plead the Fifth.
It is worth noting that the right against self-incrimination only extends to speech and not to other information that might lead to self-incrimination. A suspect taken into police custody may not refuse to be fingerprinted, nor may he refuse a DNA test or anything of the sort by pleading the Fifth. The Fifth Amendment guarantees the right to remain silent, not the right to immunity.