Many programs contain C in them but contain predominantly other code. 

What parts of a program should be written in C, what are these low level-parts?

See AlternateHardAndSoftLayers for some discussion.

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The quick answer is, when you need it to run faster, profile the program, and make sure that the slowest parts are using optimal algorithms. Then, if it's still too slow, recode the slowest parts in C.

Also sometimes C is used to interface directly to the OS and/or hardware, when the main language doesn't do so effectively (e.g. many languages don't support pointers nor system calls directly).

Sometimes the CPU has capabilities (instructions) that are not available even from C, such as saturating arithmetic. In these cases, we may code parts in assembly.
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C is sometimes called the "modern-day assembly language".
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JotEngine in PythonLanguage has some parts written in CeeLanguage. 
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See Also PaveThePath
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CategoryPattern CategoryCee