How do NPN bipolar junction transistors turn on and off?

In order to bias a bipolar junction transistor on, you need to forward bias the base-emitter junction at the same time you forward bias the collector-emitter junction, and the ratio of collector current over base current must be somewhat less than hFe, the transistor's gain. This is known as saturated, or non-linear mode, operation. In practice, we drive the base much harder than the calculated required current, so as to minimize dependency on varying hFe's for various transistors.

Turning the transistor off is a simple matter of eliminating the base current.

In the case of the NPN transistor, the base and collector would need to be more positive than the emitter. In the case of the PNP, they would need to be more negative.