The points
Under Rule 21.1 of the Rules of Golf, each hole scores against a fixed target (par, or net par in a handicap competition): two or more over scores 0 points, one over scores 1, the target scores 2, one under scores 3, two under scores 4, and three under scores 5. Add the 18 holes together; the highest total wins.
Why groups like it
A 9 on one hole costs at most two points against a bogey, and you can pick up once you can't score — so rounds move faster and a bad start doesn't end anyone's day. That's also why it suits groups with a wide spread of handicaps.
Net Stableford
In a handicap Stableford your strokes are given on the holes the scorecard rates hardest (the stroke index). A golfer getting a stroke on a par 4 scores a net par with a 5, worth 2 points. The recommended allowance for individual Stableford is 95% of the course handicap.
Modified Stableford
Some events use a version that rewards aggressive golf — for example plus points for birdies and eagles and minus points for bogeys or worse. It's fine for a group, as long as everyone knows the table before they tee off.
What it needs
Points are worked out hole by hole, so a Stableford competition needs a score for every hole — a single total for the round can't be scored.