the ar15 uses a gas operating system right?  what are all the main operating systems used by assault rifles?  how do they compare to each other?

There's the direct gas system used by the AR where some gas is redirected back to the bolt carrier via a narrow tube.

There's the gas piston systems where some gas is diverted into a large chamber containing a piston or operating rod that connects back to the bolt carrier.  Pretty much every gas operated rifle except the AR uses some variation of this.

There's the simple blowback, but that only works well with pistol calibers.

There's the delayed blowback system used by HK and CETME that uses a roller system to keep it locked up.

There's electrically operated, but I don't know of any that have ever actually been produced in any numbers.