Greek for "cross examination"
Socrates used this method to elicit truth by grilling his opponents in a cross examination to determine truth by agreeing on certain definitions and principles from the outset, and using reason to draw out the shape of the truth from the flow of conversation.
This is also sometimes referred to as the 'dialectic' method, but the dialectic is a much more analytical and rhetorically sharp approach. Elenchus has an open feel, the cross-examination, questioning, while hard, is not supposed to be totally destructive or devastating, but rather and humble and educational for both sides.
This is the perfect platform from which to launch into Socratic Irony, though, so be warned.