What is the name of the GUI Android uses/has used? Like the GUI controlling home screen and app icons, etc.
How or what is this called that renders this?
It gets the name of the Android OEM front end interface :


Touchwiz for Samsung
HTC Sense for HTC
MotoBlur for Motorola
...


For the "basic" version of Android - i.e. the version embedded in Nexus phones and tablets - people often refer to the Android version as "stock Android". So what about "stock Android UI" or "stock UI" in this case ?