The argument from silence (also called argumentum a silentio in Latin) is that the silence of a speaker or writer about X proves or suggests that the speaker or writer is ignorant of X. Whether such an argument is reasonable is subject to some interpretation or debate -- in general, the argument from silence does not offer a rigorous logical proof of a premise, although it may potentially offer some circumstantial evidence for a position.
Here is an easily recognizable example:
Bobby: I know where you live.
Bobby: I'm not telling you!
Billy: You're just saying that because you don't know!
Obviously, the Argument from Silence has limits. Wikipedia offers this logically parallel, but unconvincing, argument:
John: Do you know your wife's email password?
Jack: Yes, I do as a matter of fact.
John: What is it?
Jack: Hey, that's none of your business.
So when does the Argument from Silence work? Signaling models can light the way. The Argument from Silence is probative (though not decisive!) if, on average, people who know are better off speaking, and people who don't know are better off silent.
On exams, for example, we reason that if a student knows the answer to a question, he will write it down, and if he doesn't know the answer, he won't. So we infer ignorance from silence. In contrast, if a student fails to mention Tolstoy on a math exam, we don't infer ignorance of Tolstoy, because talking about Tolstoy on a math exam takes time and won't raise your grade.
An obvious extension: In a trial, does the failure to take the stand in one's own defense indicate guilt? While you could concoct a elaborate story about why an innocent man would be worse off telling his side of things, it wouldn't be very plausible. Silence is, probabilistically, an admission of guilt, whatever the law says.
But can you really believe what I've just written? After all, I have an incentive to say it even if I don't believe it - it might get me get out of jury duty one day!