If you have a manual-transmission car, you already know the answer: The clutch does--you have to push the pedal in so the engine and transmission are no longer connected together.
Automatic transmissions have a liquid connection between the engine and transmission: The torque converter. It's a little like having a fan that blows air at another fan, and the air turns the second fan's blades. That's a vast oversimplification of this one aspect of the torque converter's function, but hopefully it gives you the idea.
With an automatic, there is no solid connection between the engine and the transmission. (Though most cars now have "lockup torque converters" that do make a solid connection, that generally only happens at higher speeds. At low speeds, there is no solid connection.)
Here is lots and lots of reading for you:
HowStuffWorks "How Automobiles Work"
For the transmission, see:
HowStuffWorks "How Automatic Transmissions Work"
-soD