Is Python interpreted, or compiled, or both?

From my understanding:

An interpreted language is a high-level language run and executed by an interpreter (a program which converts the high-level language to machine code and then executing) on the go; it processes the program a little at a time.

A compiled language is a high-level language whose code is first converted to machine-code by a compiler (a program which converts the high-level language to machine code) and then executed by an executor (another program for running the code).

Correct me if my definitions are wrong.

Now coming back to Python, I am bit confused about this. Everywhere you learn that Python is an interpreted language, but it's interpreted to some intermediate code (like byte-code or IL) and not to the machine code. So which program then executes the IM code? Please help me understand how a Python script is handled and run.

First off, interpreted/compiled is not a property of the language but a property of the implementation. For most languages, most if not all implementations fall in one category, so one might save a few words saying the language is interpreted/compiled too, but it's still an important distinction, both because it aids understanding and because there are quite a few languages with usable implementations of both kinds (mostly in the realm of functional languages, see Haskell and ML). In addition, there are C interpreters and projects that attempt to compile a subset of Python to C or C++ code (and subsequently to machine code).

Second, compilation is not restricted to ahead-of-time compilation to native machine code. A compiler is, more generally, a program that converts a program in one programming language into a program in another programming language (arguably, you can even have a compiler with the same input and output language if significant transformations are applied). And JIT compilers compile to native machine code at runtime, which can give speed very close to or even better than ahead of time compilation (depending on the benchmark and the quality of the implementations compared).

But to stop nitpicking and answer the question you meant to ask: Practically (read: using a somewhat popular and mature implementation), Python is compiled. Not compiled to machine code ahead of time (i.e. "compiled" by the restricted and wrong, but alas common definition), "only" compiled to bytecode, but it's still compilation with at least some of the benefits. For example, the statement a = b.c() is compiled to a byte stream which, when "disassembled", looks somewhat like load 0 (b); load_str 'c'; get_attr; call_function 0; store 1 (a). This is a simplification, it's actually less readable and a bit more low-level - you can experiment with the standard library dis module and see what the real deal looks like. Interpreting this is faster than interpreting from a higher-level representation.

That bytecode is either interpreted (note that there's a difference, both in theory and in practical performance, between interpreting directly and first compiling to some intermediate representation and interpret that), as with the reference implementation (CPython), or both interpreted and compiled to optimized machine code at runtime, as with PyPy.

The CPU can only understand machine code indeed. For interpreted programs, the ultimate goal of an interpreter is to "interpret" the program code into machine code. However, usually a modern interpreted language does not interpret human code directly because it is too inefficient.

The Python interpreter first reads the human code and optimizes it to some intermediate code before interpreting it into machine code. That's why you always need another program to run a Python script, unlike in C++ where you can run the compiled executable of your code directly. For example, c:\Python27\python.exe or /usr/bin/python.