Transistors function as switches and amplifiers by operating in different regions: cut-off and saturation for switching, and the active region for amplification. In cut-off, no current flows, while in saturation, a large current flows, effectively turning the switch "ON." Amplifiers typically handle small signals, as larger signals can exceed supply limits, leading to distortion. The common emitter configuration is the most common for amplifying signals, where the input is applied to the base, affecting the collector current. Understanding the transistor's operation in these modes is crucial for effective circuit design.