Nearly all PGA Tour tournaments involve 72 holes of golf played over four days. The field typically starts with between 132 and 156 players. After the first two days of play (36 holes), the field is cut about in half; golfers with scores in the weaker half are eliminated from competition and don't play the final two days. In many tournaments not making the cut means not getting paid. Typically the top 70 players in the field and ties make the cut.
In 2008, the PGA Tour Policy Board changed the rules regarding the cut, adding a designation called MDF, which stands for "Made (Cut) Didn't Finish." It remained true that the top 70 golfers and any ties still made the cut. But if the number of golfers who made the cut was more than 78, only golfers with the score nearest to 70 people played the final two days of the tournament. People who made the cut yet didn't continue on to the final two rounds are given the MDF designation for that event and paid.
A specific example from an ESPN.com article in 2008 makes this easier to understand. At that year's Sony Open, after 36 holes of play there were 87 players who finished at even par. Before the MDF rule all of them would have both made the cut and continued play. But because the number was greater than 78, some of the golfers were given the MDF designation. There were 69 players who were 1-under par or better. Since that was the nearest number of players to 70, only those 69 played the final two days. The other 18 players were given the MDF.
Why the Change
The rules were changed to ensure that the size of the fields was manageable. Having fewer people playing down the stretch in tournaments speeds up play and allows more flexibility in the event of inclement weather. There were events pre-MDF in which 80 or 90 golfers made the cut, which the PGA felt was too many. Television also favored the change, as it prevented Monday finishes that get lower ratings.