System calls are functions that a programmer can call to perform the services of the operating system. if you want to see which system calls a program uses, run strace
In general, a process is not supposed to be able to access the kernel. It can't access kernel memory and it can't call kernel functions. The hardware of the CPU enforces this (that's the reason why it's called `protected mode').
System calls are an exception to this general rule. What happens is that the process fills the registers with the appropriate values and then calls a special instruction which jumps to a previously defined location in the kernel (of course, that location is readable by user processes, it is not writable by them). Under Intel CPUs, this is done by means of interrupt 0x80. The hardware knows that once you jump to this location, you are no longer running in restricted user mode, but as the operating system kernel --- and therefore you're allowed to do whatever you want.
The location in the kernel a process can jump to is called system_call. The procedure at that location checks the system call number, which tells the kernel what service the process requested. Then, it looks at the table of system calls (sys_call_table) to see the address of the kernel function to call. Then it calls the function, and after it returns, does a few system checks and then return back to the process (or to a different process, if the process time ran out). If you want to read this code, it's at the source file arch/$<$architecture$>$/kernel/entry.S, after the line ENTRY(system_call).
So, if we want to change the way a certain system call works, what we need to do is to write our own function to implement it (usually by adding a bit of our own code, and then calling the original function) and then change the pointer at sys_call_table to point to our function. Because we might be removed later and we don't want to leave the system in an unstable state, it's important for cleanup_module to restore the table to its original state.
Links:
Linux System Call Quick Reference -
System Calls - http://www.faqs.org/docs/kernel/x931.html