Q:

What is the C runtime library? 

What actually is a C runtime library and what is it used for?  I was searching, Googling like a devil, but I couldn't find anything better than Microsoft's: "The Microsoft run-time library provides routines for programming for the Microsoft Windows operating system.  These routines automate many common programming tasks that are not provided by the C and C++ languages." 
OK, I get that, but for example, what is in libcmt.lib?  What does it do?  I thought that the C standard library was a part of C compiler.  So is libcmt.lib Windows' implementation of C standard library functions to work under win32? 

A:

Yes, libcmt is (one of several) implementations of the C standard library provided with Microsoft's compiler.  They provide both "debug" and "release" versions of three basic types of libraries: single-threaded (always statically linked), multi-threaded statically linked, and multi-threaded dynamically linked (though, depending on the compiler version you're using, some of those may not be present). 
So, in the name "libcmt", "libc" is the (more or less) traditional name for the C library.  The "mt" means "multi-threaded".  A "debug" version would have a "d" added to the end, giving "libcmtd". 
As far as what functions it includes, the C standard (part 7, if you happen to care) defines a set of functions a conforming (hosted) implementation must supply.  Most vendors (including Microsoft) add various other functions themselves (for compatibility, to provide capabilities the standard functions don't address, etc.)  In most cases, it will also contain quite a few "internal" functions that are used by the compiler but not normally by the end user. 
If you want to get a complete list of the functions in "libcmt" (to use your example) you could open one of the Visual Studio command prompts (under "Visual Studio Tools", normally), switch to the directory where your libraries were installed, and type something like: lib -list libcmt.lib and it'll generate a (long) list of the names of all the object files in that library.  Those don't always correspond directly to the names of the functions, but will generally give an idea.  If you want to look at a particular object file, you can use lib -extract to extract one of those object files, then use dumpbin /symbols <object file name> to find what function(s) is/are in that particular object file. 

A:

At first, we should understand what a Runtime Library is; and think what it could mean by "Microsoft C Runtime Library". 
see: http://en.wikipedia.org/wiki/Runtime_library
I have posted most of the article here because it might get updated. 

When the source code of a computer program is translated into the respective target language by a compiler, it would cause an extreme enlargement of program code if each command in the program and every call to a built-in function would cause the in-place generation of the complete respective program code in the target language every time.  Instead the compiler often uses compiler-specific auxiliary functions in the runtime library that are mostly not accessible to application programmers.  Depending on the compiler manufacturer, the runtime library will sometimes also contain the standard library of the respective compiler or be contained in it. 
Also some functions that can be performed only (or are more efficient or accurate) at runtime are implemented in the runtime library, e.g. some logic errors, array bounds checking, dynamic type checking, exception handling and possibly debugging functionality.  For this reason, some programming bugs are not discovered until the program is tested in a "live" environment with real data, despite sophisticated compile-time checking and pre-release testing.  In this case, the end user may encounter a runtime error message. 
Usually the runtime library realizes many functions by accessing the operating system.  Many programming languages have built-in functions that do not necessarily have to be realized in the compiler, but can be implemented in the runtime library.  So the border between runtime library and standard library is up to the compiler manufacturer.  Therefore a runtime library is always compiler-specific and platform-specific. 
The concept of a runtime library should not be confused with an ordinary program library like that created by an application programmer or delivered by a third party or a dynamic library, meaning a program library linked at run time.  For example, the programming language C requires only a minimal runtime library (commonly called crt0) but defines a large standard library (called C standard library) that each implementation has to deliver. 

A:

I just asked this myself and was hurting my brain for some hours.  Still did not find anything that really makes a point.  Everybody that does write something to a topic is not able to actually "teach".  If you want to teach someone, take the most basic language a person understands, so he does not need to care about other topics when handling a topic.  So I came to a conclusion for myself that seems to fit well in all this chaos. 
In the programming language C, every program starts with the main() function. 
Other languages might define other functions where the program starts.  But a processor does not know the main().  A processor knows only predefined commands, represented by combinations of 0 and 1. 
In microprocessor programming, not having an underlying operating system (Microsoft Windows, Linux, MacOS,..), you need to tell the processor explicitly where to start by setting the ProgramCounter (PC) that iterates and jumps (loops, function calls) within the commands known to the processor.  You need to know how big the RAM is, you need to set the position of the program stack (local variables), as well as the position of the heap (dynamic variables) and the location of global variables (I guess it was called SSA?)  within the RAM. 
A single processor can only execute one program at a time. 
That's where the operating system comes in.  The operating system itself is a program that runs on the processor.  A program that allows the execution of custom code.  Runs multiple programs at a time by switching between the execution codes of the programs (which are loaded into the RAM).  But the operating system IS A PROGRAM, each program is written differently.  Simply putting the code of your custom program into RAM will not run it, the operating system does not know it.  You need to call functions on the operating system that registers your program, tell the operating system how much memory the program needs, where the entry point into the program is located (the main() function in case of C).  And this is what I guess is located within the Runtime Library, and explains why you need a special library for each operating system, cause these are just programs themselves and have different functions to do these things. 
This also explains why it is NOT dynamically linked at runtime as .dll files are, even if it is called a RUNTIME Library.  The Runtime Library needs to be linked statically, because it is needed at startup of your program.  The Runtime Library injects/connects your custom program into/to another program (the operating system) at RUNTIME.  This really causes some brain f...
Conclusion:
RUNTIME Library is a fail in naming.  There might not have been a .dll (linking at runtime) in the early times and the issue of understanding the difference simply did not exist.  But even if this is true, the name is badly chosen. 
Better names for the Runtime Library could be: StartupLibrary/OSEntryLibrary/SystemConnectLibrary/OSConnectLibrary
Hope I got it right, up for correction/expansion. 
cheers.