Transistors are analogous to water faucet, except that instead of water, they control the flow of electrons, or current. Not only do they start and stop the flow of a current, but they also control the amount of the current.
Transistors are used in all the technological wonders of our everyday life: cell phones, computers, cars, etc. Before transistors, vacuum tubes and electromechanical switches were used to complete electrical circuits, which were not ideal, since they were bulky, had to be warmed up, and used way too much energy. In the1940's, the Bell Telephone Labs assembled an all-star team of scientific minds, including John Bardeen, Walter Brattain and William Shockley, and to research on vacuum tube substitutes. In 1948, Bell Labs announced to the world that it had invented working transistors, which later earned them a Nobel Prize in 1956. On the right is the dream team, and below is the stylized replica of the first transistor!

MOSFET or Metal-Oxide-Semiconductor Field-Effect- Transistor is a transistor used for amplifying or switching electronic signals. It has 3 terminals labelled "Gate", "Source", and "Drain". In enhancement mode MOSFETs, a voltage drop across the oxide, or the Gate, induces a conducting channel between the Source and Drain contacts via the field effect. The term "enhancement mode" refers to the increase of conductivity with increase in oxide field that adds carriers to the channel. The channel can contain electrons (called an nMOSFET or nMOS), or holes (called a pMOSFET or pMOS), opposite in type to the substrate, so nMOS is made with a p-type substrate, and pMOS with an n-type substrate. So the increasing conductivity allows electrons to flow through the channel, completing the circuit. Hence, the MOSFET is a voltage-controlled resistor - the voltage at the Gate determines whether the channel will be in conducting or in non-conducting state - acting like a valve.
The possibility of making MOSFETs very small, down to ~ 100 nm, has revolutionized technology, by giving the power to make smaller chips. In this video, Adam Savage and Jamie Hyneman show how Intel's incredible shrinking transistors are helping to cram supercomputer performance into small laptops!
Using MOSFET to control LEGO motor:

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