Clone vs Counterfeit (Arduino)
This is not gospel, but here is what those terms are generally used to indicate (as I understand it):


compatible: a dev board which is compatible with the Arduino IDE/toolchain. The Arduino UNO uses an ATmega328 MCU, while other platforms may use different MCUs i.e. there are multiple MCUs that are compatible with the Arduino IDE. It may or may not have the same form factor. Keep in mind that this term can also be used for shields, "Arduino-compatible" shields are add-on boards that are meant to mate with an Arduino. The Sparkfun RedBoard is an example.
clone: a dev board which has the same (or very similar) circuitry as an Arduino and the same physical board/layout. They are branded differently, but are, for all intents and purposes, exactly the same as a real Arduino. These boards are permitted under the open-source licensing of the Arduino design. An Freeduino might be an example.
counterfeit: a dev board which is identical to an original Arduino, and is branded and sold as an original Arduino. As I understand it, the open-source licensing does not permit this, however, this does not deter the flood of shops churning out counterfeit Arduinos. A search on eBay or Aliexpress will turn up loads of examples; note the Arduino guide to spotting counterfeits, in general, the silkscreen is blue and the quality of assembly is not quite as good.


The difference between "compatible" and "clone" is a bit blurry, but it's not really important. The main difference between counterfeit and non-counterfeit is that counterfeit Arduinos are boards that are not sold by Arduino but are branded as such.