Stableford 
Stableford is a points based system where you earn points for your score on each hole, the higher the points total, the better. Any score worse than a nett double bogey receives 0pts. 1pt for a nett Bogey, 2pts for a nett Par, 3pts for a nett Birdie, 4pts for a nett Eagle & 5pts for a nett Albatross.
Modified Stableford 
Similar to traditional Stableford, but you are able to select the value of points awarded for each score. E.g. Changing the value of a nett Par from 2pts to 4pts.
Strokeplay 
Stroke play is a format where a golfer counts the strokes used to complete each hole, the total number of strokes on these holes is added up at the end creating a round total.
Matchplay 
Matchplay is when the player or players (if playing pairs or foursomes / greensomes) with the lowest score wins the hole. The Player that wins more holes wins the match. If you play with handicaps, the difference is taken between the handicaps, the player with the lowest handicap plays off 0 and the higher handicap plays of the difference.
Example
Player 1 - Hcap 10 - Shots 0
Player 2 - Hcap 18 - Shots 8