These sensors use a solid-state technique to determine the temperature. That is to say, they don't use mercury (like old thermometers), bimetalic strips (like in some home thermometers or stoves), nor do they use thermistors (temperature sensitive resistors). Instead, they use the fact as temperature increases, the voltage across a diode increases at a known rate. (Technically, this is actually the voltage drop between the base and emitter - the Vbe - of a transistor.) By precisely amplifying the voltage change, it is easy to generate an analog signal that is directly proportional to temperature. There have been some improvements on the technique but, essentially that is how temperature is measured.
The good news is all that complex calculation is done inside the chip - it just spits out the temperature, ready for you to use!
Because these sensors have no moving parts, they are precise, never wear out, don't need calibration, work under many environmental conditions, and are consistant between sensors and readings. Moreover they are very inexpensive and quite easy to use.
These stats are for the temperature sensor in the Adafruit shop, the Analog Devices TMP36 (-40 to 150C). Its very similar to the LM35/TMP35 (Celsius output) and LM34/TMP34 (Farenheit output). The reason we went with the '36 instead of the '35 or '34 is that this sensor has a very wide range and doesn't require a negative voltage to read sub-zero temperatures. Otherwise, the functionality is basically the same.
To convert the voltage to temperature, simply use the basic formula:
Temp in °C = [(Vout in mV) - 500] / 10
So for example, if the voltage out is 1V that means that the temperature is ((1000 mV - 500) / 10) = 50 °C
If you're using a LM35 or similar, use line 'a' in the image above and the formula: Temp in °C = (Vout in mV) / 10