With the Golf Handicap Calculator (currently available only for android devices) from earth42, you can enter rating and slope parameters of golf courses you play, and then enter your score and course for each game. Once you have at least 5 games stored, it will calculate and graph your "differentials", the normalized scores, and your "handicap index"; and once you have a handicap index, the calculator can also predict your score for a stored course, or from a rating and slope that you enter.
NEWSFLASH: I have recently had a very helpful conversation with theoclitus, a user who plays nine-hole golf courses. These courses have nine-hole ratings, and the newest version, 3.0, deals with both nine-hole and eighteen-hole courses - just register the course in your list with its nine-hole rating and slope, and check the box indicating it's a nine-hole course. If you play nine on an eighteen-hole course, and they don't furnish nine-hole ratings and slopes for the front and back, you can just use the course's given rating and slope, but you must double your score when you enter the game in order to get the correct handicap index, or at least as close as you can get. There is now also a preference in Settings to choose whether to display your handicap index as eighteen-hole or nine-hole, and regardless of that, you can select nine-hole or not when predicting your score.
To add more courses, click the add option from the options menu.
While it is not necessary to calculate your handicap index, you should add your games in chronological order. Then the graphic on the home page will illustrate your wonderfully improving trend :-)
When you have twenty games stored, the Add a Game function will automatically roll the oldest game off the list when it adds your new game. Under USGA rules, the most recent twenty games are used to calculate your handicap index, though you only need five to start with.