The flywheel (if manual) or flex plate (if automatic transmission) is a disc that is bolted to the crankshaft at the rear of the engine. It serves several purposes:
On a manual transmission vehicle, it provides one of the friction surfaces for the clutch. The clutch pressure plate is bolted to it and the clutch disk is sandwiched between the two.
On an automatic transmission vehicle it is actually called a flex plate and the torque converter of the transmission bolts to it.
It also has a toothed ring on its outer edge that the starter turns to start the engine. Because the disc is large (relative to the crankshaft) this allows the starter motor to have more leverage against the engine.