This is a really complex topic, but I'll try to address the two most popular versions:

Resistive screens use two flexible layers. When the user presses, the two layers come in contact with each other completing a circuit. Resistance varies with distance. The system determines how far along the x or y axis the press is by measuring the resistance in the circuit. This is the most common type of touchscreen and is used on the Nintendo DS.

The Apple iPhone uses a different technology: capacitive touch. The screen has a known electrical capacitance (stored energy). Contact with the human body causes this capacitance to change (due to our capacitance). The screen detects these changes to track finger touches.

Capacitive screens cost more to manufacture, but are more robust because they don't have flexible parts. They can also track multiple touches, a resistive screen can't.

These two touchscreen technologies should not be confused with the multitouch interaction exhibited by Microsoft Surface. Surface uses IR light and a camera to track multiple points. It's not a thin-film technology like normal touchscreens.
Source(s):
http://en.wikipedia.org/wiki/Touchscreen
http://en.wikipedia.org/wiki/Multi-touch


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