Now to game elements. According to Prof. Kevin Werbach, game elements are regular patterns that appear in any game. They can be extracted from a game and applied to a gamified system. When you try to create a gamified system by introducing game elements to it, what you are trying to produce is a certain kind of experience which is not the same as the game itself. The best example of a gamified system would not be the one that uses the most elements but the one that uses them most effectively.
Prof. K. Werbach has developed a framework for gamification elements in a form of pyramid which can be seen on the screenshot of the professor's video lecture. The notion of the pyramid is that we have a few dynamics, a larger number of mechanics and even a bigger number of components.
At the next level are the game mechanics. These are the elements that move the action forward. Some of the game mechanics are challenges - some goal to reach; chance - something that makes the result random; cooperation and competition to have the notion of winning and losing; feedback - very important for players to see how they are doing as this drives them to continue; resource acquisitions - things like rewards, things that can be sold/bought or exchanged.
Finally, game components are at the lowest level of the pyramid. The examples of the lowest level are examples of elements higher up. So, for example, achievements, as opposed to the general idea of challenge, giving the player some rewards attached to doing a set of specific tasks - that is the achievement; or avatars/badges - specific visual representations of those achievements, etc.
There are some game elements that are more common than others and are more influential than others in shaping typical examples of gamification. These are represented by the acronym PBL - points, badges and leaderboards. However important these three are, gamification should never start and end with these three only.
Points are a way of determining how well someone is doing in the game. They can show the relative position of one player against the others or define winning. They can connect up with rewards as well as provide feedback and/or display progress.
Badges are visual representations of one's achievements/level. They are button-like graphic that go on a profile page or some place where other players can see them. Badges can represent anything and motivate a certain behaviour. Badges also function as credentials and they are a status symbol. (For more information about badges read here).
Leaderboards are about ranking. The feedback these provide is on competition. Not to disappoint their players a number of social games have introduced personalized leaderboards where players see their name in the middle not at the bottom of the leaderboard. Another option of this is friend leaderboards where one competes against their friends not against complete strangers. Think of the leaderboard in FarmVille on Facebook for an example of this. However, learderboards can also demotivate or make them less willing to engage because of the stress they put on players.
But game elements are only a starting point for gamification. By just randomly throwing them into a gamified system, one won't be able to make it fun and engaging. What makes the elements successful is the way they are put together and that often involves resources some of the higher level. If there is a heavy emphasis on PBLs, rewards may be overemphasized. Rewards are not the same as fun; they aren't wrong but if they the only thing then there is a danger that the system will not generate the necessary level of engagement and may not actually draw in people with different kinds of motivation.
Further Reading - MDA Formal Approach to Game Design