A game system is a mostly self-contained system within a game, but is usually not a very interesting game in and of itself. Interesting game behaviour generally arises from the interaction of multiple systems. For instance, a First-Person Shooter, like Doom contains:
The first four of these are obviously essential to making an interesting game, the next three add depth to the game, and the automap is a useful player tool.
Why Analyze A Game As Interacting Game Systems?
For Game Developers: Almost no game is a complete success or abject failure. Almost every game has at least one successful idea or implementation that can be taken and applied to other games, and almost every game has at least one bad idea or implementation that can be avoided in other games. It can be helpful to isolate these ideas/implementations to particular game systems or an interaction between a particular set of game systems.
For Game Players: Some people are really picky about how a particular system is implemented. I know several people who get turned off by games that have bad Third Person Camera systems. Others may not want to deal with Resource Management. By analyzing a game in terms of what game systems it implements, and how well it implements them, a prospective player can have a better idea about whether or not he or she wants to invest the time and money in playing the game.
Other Common Systems(and some examples)
Of course, there are many more systems than that, but most tend to be either less frequently used or more specialized versions of general systems. Some, of course, I've simply overlooked.