Early transistors were made with germanium but silicon is used almost universally in semiconductor manufacturing. A single transistor is a few millimetres wide and tall and has three terminals. It works by transferring a high resistance input to a lower resistance output. The transfer of resistance gave rise to the name trans - istor. A transistor will generate a large change in current for a given input or a large change in voltage. There are numerous ways to use them and the applications and science of the transistor can fill many books - too complex to discuss in detail in this answer.
Transistors are good amplifiers and are found in radios, music equipment, video systems in their role as an amplifier. They are also very efficient switches and because they will switch from low to high, are common in binary circuits. The transistor is the basis of all digital computers. A modern CPU chip in a computer is actually made of many thousands of transistors inside the integrated circuit. A typical computer will have millions of transistors in total.
For further reading, take a look for basic electronics tutorials. All of them will have detailed descriptions of the way a transistor works and demonstrate the more common applications.