What is saturation and active region in a transistor?

The output current of a transistor is controlled by the current in the 'base' input: Increasing the control current will increase the output current in a more or less linear fashion.

In the saturation region, this is no longer true: The transistor is nearing the limits of how much current it can conduct, so increasing the control current further has little or no effect.

When using a transistor as an amplifier, you want to stay away from the saturation region as it would distort the signal you are amplifying.

When using a transistor as an on/off switch, as in digital circuits, being in the saturated region is 'on' and a normal mode of operation.