There are two kinds, Pressure Sensitive and Capacitance ones. Pressure Sensitive is less sensitive, but simpler to make and understand. It has a bunch of wires that have current going through them sandwiched between 2 plastic sheets. The wires form a grid with some current wires going up/down and others left/right, like on a chess board. When you press at a point, such as using a stylus pen, it breaks the current of one of the wires and a sensor can tell where it happened by looking at the X and Y coordinates.
The next kinds is more complicated and is what most modern touch screens use, but the concept is similar. They use capacitance and not pressure. It too has a grid of wires, but instead of sensing breaks in current when pressed at a point, it measures voltage changes. Your fingers can conduct charge and is the reason you sometimes get a shock when you drag your feet on the floor and touch something. The voltage changes when your finger is near the electric wires since your finger starts to conduct some of the charge. It is a very tiny charge, so you don't feel it. Sensors at each of the four corners of the touch screen are used to determine the location and even what type of touch it was, like a swipe, or hard press.