The compileSdkVersion is the version of the API the app is compiled against. This means you can use Android API features included in that version of the API (as well as all previous versions, obviously). If you try and use API 16 features but set compileSdkVersion to 15, you will get a compilation error. If you set compileSdkVersion to 16 you can still run the app on a API 15 device as long as your app's execution paths do not attempt to invoke any APIs specific to API 16.
The targetSdkVersion has nothing to do with how your app is compiled or what APIs you can utilize. The targetSdkVersion is supposed to indicate that you have tested your app on (presumably up to and including) the version you specify. This is more like a certification or sign off you are giving the Android OS as a hint to how it should handle your app in terms of OS features.
