So I spent a lot of time trying to fugure out what Cython and Jython are and I'm still (more) confused as anyone who just started computer programming. I heard that Cython is an extension, but..is also and indepent language??
What I think I've understood is that:

Cython/Jython is just Python and you can use C or C++/Java libraries respectively with just a little bit of Cython/Jython syntax.
It's meant to speed up performance as well as improve code readability when a task would be more efficient to write in C or C++/Java and, this is done by using statically typed variables.

Or is Cython/Jython just some sort of extension that is used sometimes but not independently? What I mean is, you'd still write everything in Python and then, for the tasks you'd rather use C or C++/Java for, you'd use Cython/Jython instead? (I know I still need Python installed and similar things as it's dependent.)
Because if it really is better, other than the barrier that it's still being developed since it's rather new, wouldn't the need for C or C++/Java completely go away as these are as easy as Python but as powerful as C or C++/Java?
CPython is comparable to Jython. They're the implementation of the Python language. CPython is the de-facto standard and was written in C. Jython is written in Java and runs on the JVM. It also allows accessing the Java ecosystem to a great extent. There are also other implementation of the language, like PyPy or Pyston. 
Cython is totally different. It allows us to write extensions for Python in C or Pyrex, a subset of the Python language. Cython speeds up the execution speed for the parts written with it.
Is Cython/Jython an independent language?