Factorial Function in C

I recently started reading Hacking: The Art of Exploitation by Jon Erickson. In the book he has the following function, which I will refer to as (A):

int factorial (int x)       // .......... 1
{                           // .......... 2
    int i;                  // .......... 3
    for (i = 1; i < x; i++) // .......... 4
        x *= i;             // .......... 5
    return x;               // .......... 6
}                           // .......... 7

This particular function is on pg. 17. Up until this function, I have understood everything he has described. To be fair, he has explained all of the elements within (A) in detail, with the exception of the return concept. However, I just don't see how (A) is suppose to describe the process of

x! = x * (x-1) * (x-2) * (x-3)

etc which I will refer to as (B). If someone could take the time to break this down in detail I would really appreciate it. Since I am asking for your help, I will go through the elements I believe I understand in order to potentially expose elements I believe I understand but actually do not but also to help you help me make the leap from how (A) is suppose to be a representation of (B).

Ok, so here is what I believe I understand.

In line 1, in (int x), x is being assigned the type integer. What I am less sure about is whether in factorial (int x), int x is being assigned the type factorial, or if even factorial is a type.
Line 3 is simple; i is being assigned the type integer.
Line 4 I am less confident on but I think I have a decent grasp of it. I'm assuming line 4 is a while-control structure with a counter. In the first segment, the counter is referred to as i and it's initial value is established as 1. I believe the second segment of line 4, i < x, dictates that while counter i is less than x, keep looping. The third segment, i++, communicates that for every valid loop/iteration of this "while a, then b" situation, you add 1 to i.
In line 5 I believe that x *= i is suppose to be shorthand for i * x but if I didn't know that this function is suppose to explain the process of calculating a factorial, I wouldn't be able to organically explain how lines 4 and 5 are suppose to interact.

'I humbly ask for your help. For any one who helps me get over this hump, I thank you in advance. '

Solution

The program loops through integers 1 thru N-1. (Assume the input value is 'N')
Before loop starts, x=N.
After one iteration, x=N*1.
After 2 iterations, x=N*1*2.
After N-1 iterations, x=N*1*2*....(N-1).
Which is N factorial.

So N! is returned.