"the name of the father" is one of Lacan's central concepts. According to Lacan's Psychoanalytic theory the force which drives us is desire. Desire is always directed at what Lacan call the Big Other or the symbolic order. The big other or symbolic order forms when we exchange objects for words and lose their real presence. This process of symbolization of reality is what allows for communication between people. But language is an outside order which precedes us and therefore, according to Lacan, is always somewhat alien to us. The term "desire" refers to the essential sense of absence which emerges upon entering the symbolic order. The individual can never fully explain himself, he can never be fully understood and his needs fully fulfilled, he can never be in full harmony with the symbolic order and shake away the feeling on constant absence.
According to Lacan desire is directed at many objects during one's life, starting from the mother which is the initial Big Other, moving to the father which is the central figure in the process of entering the symbolic order. Later on in life the role of the Big Other will be assumed by various types of authority and social institutions (Althusser elaborated on this point).
Lacan accepts Freud basic scheme of the Oedipus Complex. But unlike Freud, who thought fear of castration to be driving force in repressing the complex, Lacan argues that the fantasy of the mother in the Oedipal phase is a fantasy about complete pleasure. These fantasies are symbolically castrated upon entering the symbolic order, with castration being the separation from the mother and complete pleasure through the prohibition on incest. Giving up these two fundamental fantasies is crucial for entering the realm of language. The repression of these fantasies is made in order to receive a cental signifier, termed by Lacan as "the name of the father". The name of the father signifies absence and repression, the lack which is the result of repressing the fantasy about the mother.
Lacan's "name of the father" is a symbolic father (which can also be a woman) which is identified with the prohibiting function. This is one of Lacan most important innovations in subject theory, the idea that the subject not only becomes a subject only upon acquiring language and entering the symbolic order, but that he himself is also constructed as language. Under "the name of the father" Lacan imagines the subject's psyche as functioning the same manner language does, for example through metonyms and metaphors.