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."
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.
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 funndamental signifier which permits significaiton to proceed normally.
This fundamental signifier both confers identity on the subject (it names him, positions him within the symbolic order) and signifies the Oedipal prohibition, the 'no' of the incest taboo.
If this signifier is foreclosed (not included in the symbolic order), the result is psychosis.
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).