Points help to keep score during the game, so that a winner can be identified. They can also determine who has completed the game and reached a win state. Points can also connect to rewards, both in game or in real-life. They are also a mechanism for providing feedback on player success, so that they know if they are playing well, and is one way for the game to communicate player progress. Finally, points can also provide data for the game designer, to determine if the game is working well.
Badges are a visible indication that you have achieved something in the game. Badges are powerful because they are flexible, since the badge is simply an image that can be triggered for anything. Style is important to badges, as well. They can represent the overall aesthetic of the system. When a badge is awarded for a particular action or achievement, it signals that it is an important task, and can provide credentials that encourage respect for those achievements. In addition, badges tie into our natural desire to collect and gather. In a gamified system that relies on social status as part of the play, badges can be provide status.
Leader boards are a way to show ranking and how you are doing compared to others who play the game. They are meant to foster competition. However, they can discourage players who are at the bottom if there is a high discrepancy between their score and the top score. This causes people to leave the game. In social games, leader boards are personalized so that your score is in the middle of a sub-set of the scores that are close to you. In other cases, leader boards can be limited to only those who are connected to you, to foster competition with the people you know.