The android framework is the set of API's that allow developers to quickly and easily write apps for android phones. It consists of tools for designing UIs like buttons, text fields, image panes, and system tools like intents (for starting other apps/activities or opening files), phone controls, media players, ect. Essentially an android app consists of Activities (programs that the user interacts with), services (programs that run in the background or provide some function to other apps), and broadcast receivers (programs that catch information important to your app). The best way to learn this system will be to go through the Google Tutorials found here
AcitivityManager and LocationManager are examples of classes found in the android sdk (the framework). I do not know of any use for these classes, as I believe they are part of the system. I have never used them, but if you wanted to learn more about them i would look at the Android API
I believe your question is a bit 3 dimensional:
A library is a code source that a developer and add to their application. It is not source code, thus the inner details are hidden to the developer. You can only access the visible (public) parts.
An API is the documentation that accompanies a library to explain how to use the library (an example of this is the Android API listed above)
So to answer your question, ActivityManager and LocationManager are neither libraries nor APIs. Rather, they are classes within the Android SDK (which is a library) that are used by either the system, or the developer (if he can find any use for them). Also, everything in android is Java, so you wont find any C/C++ libraries for android
I hope that this answer was helpful for you.