What's the purpose of using assembly language inside a C program?
In addition to what everyone said: not all CPU features are exposed to C. Sometimes, especially in driver and operating system programming, one needs to explicitly work with special registers and/or commands that are not otherwise available.
Also vector extensions.
That was especially true before the advent of compiler intrinsics. Those alleviate the need for inline assembly somewhat.
One more use case for inline assembly has to do with interfacing C with reflected languages. Specifically, assembly is all but necessary if you need to call a function when its prototype is not known at compile time. In other words, when the quantity and datatypes of that function's arguments are but runtime variables. C variadic functions and the stdarg machinery won't help you in this case - they would help you parse a stack frame, but not build one. In assembly, on the other hand, it's quite doable.
This is not an OS/driver scenario. There are at least two technologies out there - Java's JNI and COM Automation - where this is a must. In case of Automation, I'm talking about the way the COM runtime is marshaling dual interfaces using their type libraries.
I can think of a very crude C alternative to assembly for that, but it'd be ugly as sin.
Yet another use case: crash/run-time error reporting. For postmortem debugging, you'd want to capture as much of program state at the point of crash as possible (i. e. all the CPU registers), and assembly is a much better vehicle for that than C.
I've been only covering cases where assembly is necessary. Hand-optimizing for performance is covered in other answers.
There are a few, although not many, cases where hand-optimized assembly language can be made to run more efficiently than assembly language generated by C compilers from C source code. Also, for developers used to assembly language, some things can just seem easier to write in assembler.
For these cases, many C compilers allow inline assembly.
However, this is becoming increasingly rare as C compilers get better and better and producing efficient code, and most platforms put restrictions on some of the low-level type of software that is often the type of software that benefits most from being written in assembler.
In general, it is performance but performance of a very specific kind. For example, the SIMD parallel instructions of a processor might not generated by the compiler. By utilizing processor specific data formats and then issuing processor specific parallel instructions (e.g. ARM NEON or Intel SSE), very fast performance on graphics or signal processing problems can occur. Even then, some compilers allow these to be expressed in C using intrinsic functions.
While it used to be common to use assembly language inserts to hand-optimize critical functions, those days are largely done. Modern compilers are very good and modern processors have very complicated timing requirements so hand optimized code is often less optimal than expected.