What does #include actually do

In practice, the difference is in the location where the preprocessor searches for the included file.

For #include <filename> the preprocessor searches in an implementation dependent manner, normally in search directories pre-designated by the compiler/IDE. This method is normally used to include standard library header files.

For #include "filename" the preprocessor searches first in the same directory as the file containing the directive, and then follows the search path used for the #include <filename> form. This method is normally used to include programmer-defined header files.

A more complete description is available in the GCC documentation on search paths.

1. A static variable inside a function keeps its value between invocations.
2. A static global variable or a function is "seen" only in the file it's declared in

(1) is the more foreign topic if you're a newbie, so here's an example:

#include <stdio.h>

void foo()
{
    int a = 10;
    static int sa = 10;

    a += 5;
    sa += 5;

    printf("a = %d, sa = %d\n", a, sa);
}


int main()
{
    int i;

    for (i = 0; i < 10; ++i)
        foo();
}

This prints:

a = 15, sa = 15
a = 15, sa = 20
a = 15, sa = 25
a = 15, sa = 30
a = 15, sa = 35
a = 15, sa = 40
a = 15, sa = 45
a = 15, sa = 50
a = 15, sa = 55
a = 15, sa = 60

This is useful for cases where a function needs to keep some state between invocations, and you don't want to use global variables. Beware, however, this feature should be used very sparingly - it makes your code not thread-safe and harder to understand.

(2) Is used widely as an "access control" feature. If you have a .c file implementing some functionality, it usually exposes only a few "public" functions to users. The rest of its functions should be made static, so that the user won't be able to access them. This is encapsulation, a good practice.

Quoting Wikipedia:

In the C programming language, static is used with global variables and functions to set their scope to the containing file. In local variables, static is used to store the variable in the statically allocated memory instead of the automatically allocated memory. While the language does not dictate the implementation of either type of memory, statically allocated memory is typically reserved in data segment of the program at compile time, while the automatically allocated memory is normally implemented as a transient call stack.

And to answer your second question, it's not like in C#.

In C++, however, static is also used to define class attributes (shared between all objects of the same class) and methods. In C there are no classes, so this feature is irrelevant.