The Ryder Cup rules are based on match play, ie between two teams or players (the Ryder Cup has both teams and single players) where you win a hole by having the lowest score (number of strokes) on that hole. The format has changed several times in the past, but the rule today is that four foursomes (a foursome is two teams of two players each) play on the first day. On the second day of the tournament, there are four fourball matches (similar to foursomes but with each competitor playing their own ball). On the last day there are twelve single player matches. A total of 28 points can be won in the Ryder Cup.
The order of play for each pairing (team of two) is decided by each team captain, and these can't be changed unless there is an emergency.
You can learn more about the Ryder Cup here.