From the Greek "to cross-examin."
An elenchus is the examination of a set of beliefs testing for coherence.
Socrates employs elenchus in his quest to find wisdom. In Plato's Socratic dialogues we find this method employed in the following way: Socrates meets an interlocutor who claims to know something; he illicits from the interlocutor a set of beliefs related to what they claim to know; Socrates then examines the beliefs to see if they are consistent with one another.
In the Apology we see this method exemplified when Socrates crosses Meletus' claims that he (Socrates) corrupts the youth of Athens, and fails to believe in the Athenian gods.
It is important to note that the elenchus in itself does not provide support for positive claims of knowledge, only that someone has an inconsistent set of beliefs, and therefore they do not know what they claim to know.