The letters S.p.A, as in Fiat S.p.A, stand for Società per Azioni. Fiat itself is also an acronym: Fabbrica Italiana Automobili Torino.
Directly translated, this means 'society of actions' but the real meaning of Società per Azioni is 'joint stock company'. It is the Italian equivalent of 'corporation' or 'limited company'.
Other Possible Meanings for S.P.A
The acronym has other uses in the world of business, including Systems Planning and Analysis, which supports American national security programs, and are based in Alexandria, Virginia, USA. SPA could also mean Sony Pictures Animation, the company responsible for films such as Open Season and The Smurfs.
For more on the history of Fiat, watch this fim: