What's the lowdown on Sega Genesis controllers?

I've seen both 3-button and 6-button controllers at thrift stores, as well as ones with joysticks attached. Whch are original, and which are third party sticks? Can all games be played with the three button controllers? If not, which use more than three buttons?

And finally, that joystick sure looks like an Atari joystick plug. Can you use Atari sticks on a Genesis? Genesis pads on an Atari? Genesis pads on my C64 would be awesome!