The name of the Father is a symbolic formation.
When the expression "the name of the father" first appeared in Lacan’s work, in the early 1950s, it is without capital letters and refers generally to the "prohibitive role" of the "symbolic father" as the one who lays down the incest taboo in the Oedipus complex.
It is in the 'name of the father' that we must recognize the support of the symbolic function which, from the dawn of history, has identified his person with the figure of the law.
The 'No" of the Father
From the beginning Lacan plays on the homophony of le nom du père (the name of the father) and le 'non' du père (the 'no' of the father), to emphasize the legislative and prohibitive function of the symbolic father -- the 'no' of the incest taboo.
In the French language, the expression “the name of the father” (le nom du père) is phonetically similar to the expression “the ‘no’ of the father” (le ‘non’ du père). Lacan plays on this similarity to emphasize the prohibitive function of the symbolic father (the ‘no’ of the incest taboo).
The term is a play on the near-homonyms non and nom.
A few years later, in the seminar on the psychoses, the expression becomes capitalized and hyphenated and takes on a more precise meaning; the Name-of-the-Father is now the fundamental signifier which permits signification to proceed normally.
In Lacan's 1955-6 seminar, The Psychoses, the name-of-the-father is described as the fundamental signifier that both confers identity on human subjects by situating them in a lineage and the symbolic order, and reiterates the prohibition on incest.
In another work on psychosis, Lacan represents of the Oedipus complex as a metaphor (paternal metaphor), in which one signifier (the Name-of-the-Father) substitutes another (the desire of the mother).
If this signifier is foreclosed (not included in the symbolic order), the result is psychosis. Nevertheless, Jacques Lacan developed this concept with the ultimately unsuccessful aim of curing psychosis.
Nothing can be thought that is outside of language, but the phallus is there and therefore structures the whole system of thought accordingly.
Freud vs Lacan
There was a father which the brothers killed.
The brothers felt guilty about killing their father.