The purpose of dialektike, ultimately, is to replace false opinion with true knowledge by eliciting noetic insights. It aims, then, to open the eyes of the soul, or, we might also say, to activate or engage the Nous.
With Socrates and Plato, dialektike is often associated with a systematic posing and answering of questions (i.e., the 'Socratic method' or elenchus).
Dialektike is etymologically related to, but not synonymous with dialogue, a conversation between two parties. Hence we may speak of a "dialectical dialogue". Or dialectic may be pursued in a form other than dialogue.
Among the noetic insights dialectic may elicit are glimpses of the essence of virtues and Platonic Forms (Beauty, Goodness, Courage, etc.); or also logical principles and relations. Three common techniques are:
Dialektike is to be contrasted with contentious argument (eristic). Eristic is a zero-sum game (your gain is proportional to my loss); dialektike is a non-zero-sum (both win) game. With eristic, two parties are adversaries — but, with dialektike, partners. Participants in dialektike experience the felicity of camaraderie. These amiable feelings promote discovery, whereas the animosity of eristic opposes it.
An excellent technique for dialectical discussion is to have each party alternately defend and critique both sides of a particular issue.
The principles of dialectical dialogue may also be applied to one's own internal 'discussions' and deliberations.
Video from Jason Youngman (Metaphysical Reflections)