PyPy is Python implemented in Python. This means that the interpreter is written in Python, but what started as an experimental idea turned into a fast, efficient Python implementation with a JIT compiler that has the potential to implement other dynamic interpreted languages as well.

Implementation of the Python 2 specification is done in RPython – the so-called restricted Python, which is a stripped down version. The constraints that RPython has on objects make it easier to infer their types. The PyPy tool chain then has an easy time translating the RPython program to C, JVM bytecode or CIL. It also builds a JIT compiler in the resulting interpreter.

PyPy is capable of running any common Python code.

Speed

First and foremost – speed. PyPy's website maintains a page that's updated at each PyPy version which contains the accomplishments – various benchmarks and how much faster it is than those compared with CPython. These are kinda assumed,[. . .]

Well that was an astonishing achievement! The whole humanity is so proud. Here is a programmers@stackexchange thread that has some information about what its software was written in, supposedly embedded C; as well as that the tests were written in Python.