If you are actually referring to the amount of *memory* that is being used, it varies wildly depending on the application. A simpler application uses less memory than a complex one. So, for example, something like Angry Birds would probably use a lot less than a 3D action game, because less calculations are being made each second (which effects, more than anything else, the amount of memory being used.)
If you are referring to the amount of disk space used, it still varies wildly, but for different reasons. Games with lots of sounds, or games with video, will tend to take up a lot more space. Large data files can also increase it. Typically, roleplaying games and the like have much larger sizes than simpler casual games
Edit: I just noticed that this question was filed under Facebook games. In this case, the answer is almost invariably, no. Browser-based games very, very rarely use a lot of memory. If you're running windows, you can find out exactly how much memory is being used by a program at any time by pressing Control + Alt + Delete and going to your task manager. Scroll through the list to find your internet browser, and you will see a value shown for how much memory the browser is using. But I assure you, it will not be a significantly large value.