This is an old Italian card game which became popular throughout Europe in the 16th century and perhaps earlier. It is a showdown game in which players are dealt four cards. The combination which gives its name to the game is the prime, which consists of one card of each suit. This same combination appears in the Italian games Scopa and Scopone, and the card values (seven = 21, six = 18, ace = 16, five = 15, four = 14, three = 13, two = 12, pictures = 10) appear in several modern Italian games. For example in Tressette players refer to combinations of 3-2, 2-ace and 3-ace in a suit as a 25, a 28 and a 29 respectively - even though these point values have no other significance in Tressette.
Mark Waks (Justin du Coeur) has published a reconstruction of Primero based on various sources.
Another description can be found on Jeff Suzuki's Primero page.
At tretre.it, Girloamo Zorli has published articles (in Italian) on the versions of Primiera described by Berni and Cardano. The Biblioteca del tre at this site also includes a commentary on Berni's text.