Thermistor air, thermistor water & thermal fan switch.

 Thermistor air, or thermistor water are just the same name for 2 different purpose, one senses the temperature of intake air, one measures the heat of the coolant that flows in all the coolant jacket. These heats will affect the amount of voltage signal generated by these sensors, hence when sent to the ECU, it will know what temperature the air or engine is at.

  Temperature affects conductivity of materials, gradually though. Therefore the very basic principal of these thermistors are using materials that have conductivities susceptible to change of temperature. In terms of electrical, conductivity means resistance, and hence depending on different materials, resistance can rise or sink when temperature increases or decrease.

  In that case, we know have 2 well know types of thermistor: Positive Coefficient(POT) and Negative Coefficient Thermistor.
POT: Material resistance is proportional to rise of temperature & vice versa.
NOT: Material resistance is inversely proportional to rise of temperature & vice versa.

Because resistance change means voltage drop and available voltage in a fixed reference voltage circuit can be affected as well. Hence this opens up for some type of configuration of thermistor circuit. But generally, all these configurations are again, basically application of voltage divider law.
When searching for thermistor coefficients, there are 2: temperature/resistance and voltage output/resistance. In a circuit, voltage output can either be available voltage or voltage drop, as the two complement but in the same time contradict each other, therefore its definition should depend on situation.
For example:

A positive coefficient will have resistance rises as temperature rises, as resistance rises, the voltage drop across the thermistor proportionally rises as well. This configuration allows the signal to the ECU is equivalent to the voltage drop across the thermistor.

In reality, the temperature and resistance are not always proportional.

 However, this does not alter the signal going to the ECU as well as the thermistor as a whole. Because these are figures tested in boiling water environment. In real engine condition, the heat generated by combustion is ideally constant enough to keep a heating power on the coolant, thus raising the temperature rapidly. Average engine temperature varies from car to car, but hottest is always the exhaust manifold and coolant is counted as average engine temperature.

 
Coolant temperature sensor can usually be found as a plug on the side of the cylinder head, where a hole on the head is. There is 2 wires leading away one is a signal wire(red) the other one must be ground. The 5V feed is ECU internal. This makes testing the ECT simple, just by back probing the signal wire and ground the multimeter, start the engine and the temperature will change in accordance with specification, if not, you know what to do. Simplicity in design and function doesn't mean it's not important to check the ECT when the engine is starting cold or idling warm. Because, the signal makes the ECU "thinks" of how the engine is warm or cold. If the engine is warm and the ECT reads cold, the ECU will make the mixture rich and retard the spark timing, making a warm idle for your car, for example. This of course leads to bad CO and severe fuel consumption.

  As you notice, after a while, the car's fan turns on as it reaches a certain level of temperature, blowing all the really hot air in your face. That is certainly for "cooling the coolant". Because the engine temperature and the coolant's temperature balance out, the coolant can't really cool the engine so good anymore. Hence the fan will turn on to blow air temperature in-through the radiator to balance out the coolant's temperature. This whole function is packed into what we call a thermal fan switch. Sometime it is ECU controlled, most of the time it senses temperature by itself, which I think is more reliable because when other thermistor fails it will fail this thermal fan switch from the ECU.
   Thermal fan switch is kind of "backwards". The input is a mechanical change, and output is an electrical trigger. All it does is when temperature reaches a level which will eventually close the circuit of the fan's motor to battery. When the coolant decrease back to the ideal temperature, the temperature sensed @ the coolant switch material makes the contact retracted, or opened, in a few second the fan will turn off.
   Checking this switch is simple, heat it up and check for continuity, or available voltage.