What is under the hood and how different temperature sensors work?
As said in the comments, you are describing two things: Electrical interface and principle of measurement

In short for the interface:

Analog sensors are often just the sensor itself. It maps the measured property to a voltage / current / resistance / capacitance / whatever, and the "user" is responsible to interpret the output of the sensor. Some sensors may contain voltage references, amplifiers etc. but still give an analog signal.

Digital sensors can just be connected to a microcontroller, you don't need the sometimes challenging analog electronics. (I remember a capacitive humidity sensor of a few tenths of picofarad and a tiny dynamic range).

So, back to the principle of operation of temperature sensors:

Contact sensors

are those which need to touch the material to be measured There are:

PT100, PT1000,...

The simplest way is to monitor the resistance of a piece of metal. 
Platin is commonly used and the term PT100 refers to a sensor with a platin wire of 100Ohm at 0°C. The resistance only changes by a factor of 0.0038, which is not much and needs some effort to get a good precision. However, platin sensors allow for high precision as they have a very good linearity (higher order correction for more precision usually available) and are usually calibrated, i.e. each sensor has that 100Ohm at 0°C very precisely. In addition, they can be used over a large temperature rage from -250°C to 1000°C.

NTCs &amp; PTCs

As people think semiconductors can always do a better job, there are PTCs and NTCs. Their resistance becomes larger (Positive Temperature Coefficent) or smaller (Negative Temperature Coefficent) with increasing temperature. They offer a large dynamic range, the resistance is usually expressed as

$$R = R_0 \cdot \exp\left(B\left({1 \over T} - {1 \over 293.15K}\right)\right)$$

for NTCs, where Ro is thre resistance at 25°C and B a specific constant. For example: Ro=10kOhm; B=3500.

Due to this, the sensor is easy to read out and also small changes in temperature can be monitored without effort. We usually build a voltage divider of one resistor and one NTC and just connect that to an ADC.

The drawback is the nonlinearity, even the formula given above does describe an NTC over the entire usable range. There is this better suited, but ugly beast called Steinhart-Hart-Equation:

$$T=\frac{1}{ A + B \cdot \ln \left(\frac{R}{R_0}\right) + C \cdot \ln^2 \left(\frac{R}{R_0}\right) + D \cdot \ln^3 \left(\frac{R}{R_0}\right)}$$

Also, this sensors have a larger part spread (each one is a bit different, we encountered a spread of about 0.2-0.5K), are not calibrated. But they are cheap and simple, so they are widely used when absolute precision is not that important. A per-device calibration can also also increase the precision. 

Thermocouples

If you weld a wire of one metal between two pieces of a second metal, you will measure a voltage between the outer ends, which depends on the temperature difference at the welding points as well as the types of metals. So, if you know the absolute temperature of one welding point, you can determine the temperature at the other welding point. Common measurement units have the second welding point built-in together with an NTC or PT100 sensor to measure the absolute temperature.
While thermocouples exceed the temperature range of platin sensors and can be used to measure from almost 0K up to about 1800°C (using different types of couples), the voltage to me measured is tiny. A typical metal combination of NiCr and Ni gives 40µV/K... 

Usually, measurement devices have the second welding point built-in together with a PT100 or similar to get its temperature. 

Contactless sensors

measure the incident heat radiation from the object to be measured. They rely on the black-body-radiation, which is the EM-spectrum emitted by an ideal surface (a black body) for a certain spectrum:



Bolometers

Bolometers consist of an absorber which absorbs almost all EM radiation, placed on a slightly thermally insulating material. The temperature of the absorber as well as of the backside of the insulator is measured. In addition, the backside is precisely kept on a specific temperature.

So, the incident radiation turns into heat at the absorber, and the heat is transferred backwards through the insulator. The heat flow through the insulator depends on the temperature difference between front and back, so the higher the incident radiation power, the higher the temperature difference.

Having a look at the diagram above, power per square meter is the surface under a particular curve. Knowing the power means knowing the temperature. 

But you hope that the incoming radiation has the spectrum shown above. You could point this sensor into a laser beam, which would also inject lots of power, and result in a wrong temperature reading. Another issue is that the full viewing angle of the sensor has to be covered by the surface to be measured. When it's only partly covered, you'll get only a part of the radiation and a lower temperature. A solution is to use an iris for a smaller viewing angle, but you'll have to correct for the radiation power coming from the walls inside the device.

Passive infrared sensor (PIR)

Similar to the Bolometer, this device absorbs the incident radiation. The absorber is similar to a piezo crystall, which generates a charge on change of temperature instead of pressure.
As this sensors can only detect relatively fast changes of temperatures, but not slow changes and absolute temperatures, they are typically used in motion detectors.

However, I once used them to measure the intensity of a black body: the incident radiation was chopped by an optical chopper wheel and the sensor produced a triangle signal from its heating/cooling process. The slope of the "heating" part of the signal was a measure for the incident radiation power.

Two-point sensors

Have a look again at the diagram: If you know the intensity at two different wavelengths, their quotient allows to determine the shape of the full spectrum or just the temperature. One sensor for 2µm and one for 3µm would do the job.



This was quite much now, and there's still more. But it should give an overview what's available. And I think you notice that the contactless temperature sensors are not as simple as you thought.