Materials can be divided into three categories: conductors that allow electrons to flow through them, insulators that prevent the flow of electrons, and semiconductors that only let electrons flow under certain conditions. The difference between them may be best explained by the difference in their band gaps.
A band gap is an energy range in a material where no electron can exist. Conductors have no band gap, so electrons can freely move through them to generate an electric current. Metals including iron, copper, silver, gold, and aluminum are representative conductors. Insulators such as oil, glass, rubber, and ceramics have a large band gap which prevents the flow of electrons. Semiconductors, in contrast, have a small band gap, and the flow of electrons and electron holes can be controlled by adding impurities to the material.

Pure silicon and germanium crystals have insulator-like properties, and electricity hardly flows through them even when a voltage is applied. This is because their crystal lattice tightly keeps the electrons in place and hardly lets them move around.
When a very small amount of impurity such as phosphorus is introduced, however, it frees up some electrons and give the crystals conductor-like properties. Semiconductors containing impurities that produce surplus electrons are called n-type semiconductors ("n" stands for negative), and those with impurities such as boron that create a deficiency of electrons are called p-type semiconductors ("p" stands for positive). In a p-type semiconductor, electron holes rather than electrons serve as charge carriers, behaving as if positively charged electrons are flowing.
When p-type and n-type semiconductors are joined, the composite device (called p-n junction diode) produces the rectifier effect in which the flow of electric current is released or stopped depending on the direction of the electric field.

A transistor is a semiconductor device used to amplify or switch electrical signals. The name transistor is a combination of the words transfer and resistor. Transistors were developed because, once the rectifier effect had been attained with semiconductors, people needed a semiconductor device for amplifying electrical signals for telegraph and telephone.
The world's first MOS transistor was made in 1960 by Dawon Kahng and M. M. Atalla at Bell Labs. MOS transistors are the most commonly used transistors today.
They have two regions of n-type substrates separated by a wall of p-type substrate. When a positive gate voltage is applied, the top of the p-type substrate turns conductive by induction, lowering the barrier and allowing electrons to flow between the two n-type terminals. In effect, slight alterations in the gate voltage amplify changes in the output current.

A semiconductor is broadly defined today as a material with electrical conductivity that can be freely controlled by one means or another. In another words, whatever material that can be used as a transistor is a semiconductor.
There was a time when germanium and silicon were exclusively used as semiconductors, and only the group 14 elements in the periodic table were deemed to be semiconductors. As studies on compound semiconductors and organic semiconductors progressed, however, the definition of a semiconductor also changed to include all kinds of semiconducting materials, rather than just a specific group of elements.
Fairly recent additions to the category of semiconductors include carbon nanotubes discovered by Dr. Sumio Iijima, and conductive polymers discovered by Dr. Hideki Shiarakawa and others who won the Nobel Prize in Chemistry. Applications of these semiconductor materials are being studied by researchers around the world.