If a person is on trial, he does not have to implicate or incriminate himself--be a witness against himself. When a defendant takes the witness stand to defend himself, he is also subject to questioning by the prosecution. If a defendant would rather not speak on his own behalf, whatever his reason, he does not have to do so according to the self-incrimination clause of the Fifth Amendment. In pre-trial questioning, citizens are allowed to avoid self-incrimination by claiming the Fifth Amendment.
The most recent--as well as quite public and controversial--national incident of someone invoking a Fifth Amendment right against self-incrimination was the case of Lois Lerner, an Internal Revenue Service (IRS) official who appeared before a House Oversight Committee. After making a statement (which is what provided the controversy), Lerner invoked her Fifth Amendment right and did not answer any of the Committee's questions because she could not be compelled to answer.
The freedom from self-incrimination is most often referred to as "pleading the Fifth" (referring to the Fifth Amendment to the United States Constitution). This Amendment assures that American citizens, at the Federal level, are not tried for capital offenses without a Grand Jury indictment (with a few military and wartime exceptions), are not tried twice for the same offense (known as "double jeopardy"), are guaranteed due process (through the judicial system), are not deprived of their private property without appropriate compensation, or "be compelled in any criminal case to be a witness against himself." The Fourteenth Amendment grants most of these same rights to citizens at the state level, as well.