What are the advantages and disadvantages of different types of motion sensors?
Good answer @Adnan!
I am not sure what you are limiting your "motion sensor" question two, and his answer covered the common ones you find in typical systems. A few others, though to consider:
Active Fiber Optic - These typically are found in high-end perimeter barrier systems (like airport fences) and are used to detect motion and vibrations along the barrier.
Active Sonic - you have both ultra-sonic, as you noted, as well as sub-sonic. I won't get into the latter as it is a bit more exotic and only used for sensing in certain environments. These are of course active technologies - i.e., they send out a wave form and interpret it.
Active Tomographic - This is a radio wave, active sensing technology that can actually sense through walls and other obstructions. Your microwave would fall under this, and is just one type. The body scanners used at airports are even a higher frequency and are another type.
What Adnan referred to as Piezoelectric, is actually a generalized category of Vibration. These sensors can be piezoelectric, but can also be a wide range of other configurations: simple contact, spring based, ... Mercury switches are often used in these. Interestingly, due to the very small, easy to conceal, and nearly unimaginable number of configurations you can put these puppies in - these are often the hardest to take down if you are taking down a sensor. Good ones, even the slightest vibration beyond the norm for the environment create an alarm. So most often you have to identify the configuration of vibration sensors in the facility and create a vibration sensor alarm storm and get them cut off. But I digress.
Passive Electro-magnetic field sensors - These are out there, I have not run across many.
Passive chemical sensors (aka Electronic Nose) - Again these are out there, mostly in handheld units to do things like drug scans. However, they could be deployed in sensing environments to detect and screen for humans (like consider you are in a far off country hunting for a bad guy who is in the boondocks and you need to scan a trail that is used by both animals and humans).
Passive audio/acoustic sensors - these are emerging and actually are now going to be able to create 3D images of environments they sense. For instance, law enforcement can now deploy systems in hostile environments that can pinpoint to inches where a sniper is shooting from by the noise of the shot.
Passive pressure wave sensors - these could sense air pressure, water pressure, etc. As a person walks through a room they change the air flow and pressure wave and it is detectable. Again not in production environments but available.
I will also mention there are some very exotic, albeit non-production sensing technologies. I once saw a partial demonstration of what was called a Quantum Sensor. The theory was that space time has this weird 4th dimension "trail" that tapers off endlessly and with the right power equipment you can sense that trail forwards or backwards (like ripples in a pond after you drop a rock in it). The demonstration showed a very brief, but limited "hindsight", but never foresight - which of course you would need for "sensing". Of course it was very bulky equipment as well - so don't go looking for it in your average ADT security system anytime soon. But there is a lot of stuff like this that is going on all the time out there.
There are other types of motion detection methods like vibration detection, optic-based motion detection, and magnetic-based motion detection. I'm not familiar with all of these technologies, so I'll just approach the ones I've studied/played with before.
PIR (Passive Infrared): They measure the change in the energy of the surrounding area. They're prone to false positives due to being sensitive to environmental changes like hot/cold air flow and sunlight. They're easily defeated by blocking the body heat emission. If the adversary wearing a heat-insulating full-body suit or move with a styrofoam sheet, they will pass without triggering the sensor.
AIR (Active Infrared): They mainly have two implementation. Proximity sensors, used in the automatic trash bin, automatic water taps, and different others; and motion sensors. In AIR-based motion sensors, an IR emitter sends a beam of IR which will be received by an IR receiver, when the beam is interrupted, a motion is detected. Due to the way they're implemented (monitoring a specific scope) they're less prone to false positives. Their main disadvantage is detectability, they can be easily "seen" using a regular camera (your phone's camera works) or any IR detection mechanism, after that they can be easily avoided. In some cases, they might be impossible to avoid (they're monitoring the only door to a room), the adversary can detect the source of the beam and find the receiver, then emit a beam of their own. Some AIR-based motion sensors emit IR in a pattern of a certain frequency to make it difficult to replicate, but of course the adversary can learn that pattern and replay it to the receiver.
Optic-based: Basically a camera watches an area and it's recording at a certain framerate, each frame (or several frames) are analyzed by an algorithm that can detect the difference between the last frames. If something is different, a motion is detected. They can be overcome with utilizing shadows and exploiting backgrounds with a solid color. That, of course, can be solved with using thermal imaging and/or installation-specific measures (eliminating shadows with controlled light). There's a lot of cutting-edge research in this area, especially in video amplification. Researchers at MiT were able to detect a person's heart rate from a normal video footage.
Piezoelectric: I personally haven't seen or used those. They basically use the piezoelectric effect (mechanical forces having an electrical effect on some materials). They can be mounted on the floor in order to detect pressure and vibration. They're vulnerable to Flying Nimbus attacks.
Ultrasonic: They work by emitting an ultrasonic "beam". They work the same way as a sonar. They can be defeated by wearing an anechoic suit.
But to be honest, I don't think that the biggest security risk in motion-detection technology is in the sensing mechanism itself. I'd be more worried about the software/hardware controlling it, and the implementation itself.