C.F. = Codice Fiscale = Fiscal Code (for a person)
P. IVA = Partita I.V.A. (I.V.A. = V.A.T.) = the same as above, but for a society.
Studio professionale is "professional office, practice" can be translated in different ways, according to the context; for instance "consulting room" for a lawyer, "dental practice" for a dentist, etc.
Associato means associated, like in English in the same context.