The first player tees off, the second player hits the second shot, the first player hits the third shot, and so on until the ball is holed. Players alternate hitting tee shots so that the same player doesn't hit every drive.
Foursomes can be played as stroke play or match play. As match play, Foursomes is one of the formats used in the Ryder Cup.
Here's a hint for Foursomes strategy: Try to determine before the round which are the toughest driving holes on the course being played. Factor that into the decision on who hits the tee ball on the first hole. You want your best driver to be teeing off on as many of the toughest driving holes as possible.
Handicap allowances for Foursomes competitions can be found in the USGA Handicap Manual, Section 9-4 (www.usga.com).