A highly ranked person in the italian/sicilian mafia. A regime is a troop, and a capo is a leader/boss.
In The Godfather, Don Vito Corleone and Consigliori Tom Hagen is the highest, and after them comes the family's two caporegimes, Clemenza and Tessio. Later, Rocco Lampone creates a new regime.
Sonny Corleone: "Our two New York-regimes are more than a thousand men, that's more than the other five NY-mafia-families have together."
Six people were present at the meeting in Don's working-room: Michael, Sonny and Don, Tom Hagen, and the caporegimes Clemenza and Tessio.