What exactly do C include guards do?

You prevent the file from being included more than once, here

#ifndef HEADER_FILE

you test if HEADER_FILE is NOT defined, in case that's true then

#define HEADER_FILE

would define it, now if you include the file in another file, the first time it will define HEADER_FILE, while the second time, it will be already defined and hence the content of the file is not included again, since the #ifndef HEADER_FILE will be false.

Remember that these are evaluated by the preprocessor before actual compilation is done, so they are evaluated at compile time.

It's a preprocessor macro.

All of it is preprocessor syntax, that basically says, if this macro has not already been defined, define it and include all code between the #ifndef and #endif

What it accomplishes is preventing the inclusion of file more than once, which can lead to problems in your code.

Your question:

And why is it okay to forget the include guard in which case we can also forgot adding #define HEADER_FILE?

It's OK to forget it because it's still legal C code without it. The preprocessor processes your file before it's compiled and includes the specified code in your final program if there's no logic specifying why it shouldn't. It's simply a common practice, but it's not required.

A simple example might help illustrate how this works:

Your header file, header_file.h we'll say, contains this:

#ifndef HEADER_FILE
#define HEADER_FILE

int two(void){
    return 2;
}

#endif

In another file (foo.c), you might have:

#include "header_file.h"

void foo() {
    int value = two();
    printf("foo value=%d\n", value);       
}

What this will translate to once it's "preprocessed" and ready for compilation is this:

int two(void){
    return 2;
}

void foo() {
    int value = two();
    printf("foo value=%d\n", value);       
}

All the include guard is accomplishing here is determining whether or not the header contents between the #ifndef ... and #endif should be pasted in place of the original #include.

However, since that function is not declared extern or static, and is actually implemented in a header file, you'd have a problem if you tried to use it in another source file, since the function definition would not be included.