Crank Postion Sensor (CPS) - controls when the plugs fire
Cam Postion Sensor (CMP) - controls when fuel is injected. fuel injection
Note: These two are tied together so it all happens at the right time

MAF - Measures the volume of air entering the intake. Controls air/fuel mix
ICV - Not a sensor but controls idle by opening and closing to allow controlled amounts of air to bypass the throttle plate. It acts like a choke that can be adjusted by the DME.
Air Intake Temp Sensor (IAT) - Measures incoming air temp. Cold air is more dense, dense air needs more fuel
O2 Sensor - Precat - Measures exhaust gases for proper combustion. Controls air fuel mix during Closed Loop.
O2 Sensor - Postcat - Measures how well the cats are working. Doen't control anything but can indicate problems.
Coolant Temp Sensor - Measures temp of the engine. DME uses it to switch to Closed Loop at proper operating temp.
Throttle Position Sensor (TPS) - Measures changes in the throttle position. Controls RPM's.

Open Loop vs Closed Loop - 2 common terms used a lot here. Open Loop usually only happens when the car is NOT warmed up to normal operating temp. Until all the sensor readings are within their normal operating range the DME pretty much ignores the sensor readings and uses some default settings for stuff like fuel injection and air fuel ratio. Once the sensors are all reading in their normal operating range it switches to Closed Loop. Generally in Open Loop the engine is running rich. This allows it to operate smoothly and allows the engine to warm up a bit faster. In Closed Loop the DME uses the sensor readings to control spark, fuel, air, timing etc. Closed Loop is a very fine tuned computer controled environment compared to Open Loop.

Vacuum Leaks - Its not a sensor but it is a vey common problem and it absolutely contributes to a host of issues. We tell people all the time to check for vacuum leaks but few people know all the places to look or how to properly test for leaks. The best test is a Smoke Test. This uses expensive equipment to send smoke into the air intake under slight pressure. If any leaks are there the smoke will exit out the leak. The smoke test is great but you have to take it to shop to get it done. The DIY method takes a bit more time.

Start with a physical inspection. The boot between the MAF and throttle body is clamped in place. Make sure the clamps are tight. Look for cracks on the intake boot. They will most likely show up on the outside bend first. If you have a ribbed boot then look at the ribs. Cracks will be pretty visible on the top of these ribs. The best inpection of the boot is to remove it. When its installed you can only see half of it. Look at where the tube from the ICV attaches to the boot. That tube and that connection are well know for cracks and splits. Look at any tubing that attaches to the fuel rail. On M50 engines there is a small diameter tube that runs from the fuel pressure regulator to the underside of the intake maninfold. There is also a small tube at the front of the engine. These get dry and brittle over time. Check where the second tube for the ICV attaches to the back/underside of the intake manifold. Its held into the manifold with a plastic clip/connector. This can break allowing the tube to slip out. it should be located in the center of the intake directly opposite the throttle body. On M50 intake manifolds its very visible through the slots in the intake tubes. I use a screw driver to push it back in place if it comes loose. On M52 engines there are more small tubes that feed to the vacuum storage, vacuum canister, muffler flap etc. I've never worked on an M52 so Im not 100% familiar with all of them.If any tubes are cracked or broken then replace them.

To test for vacuum leaks without a smoke test get a can of carb cleaner. Start at the MAF and give each connection and each tube etc a quick shot of carb cleaner then wait a few seconds and listen for any RPM change. If you have a leak it will most likely surge a bit then settle back. work your way up the intake to the manifold. Move up and down the manifold so you can test the gaskets. Finally spray a few good shots on any other tubes you find including the brake booster tube that connects to the intake manifold.

Now for all those sensors:

Crank Position Sensor (CPS) - The crank sensor is located either at the crank pulley on the front of the engine or further back in the engine block. Its main purpose is to measure the engines rpm's and the position of the pistons. When the pistons are in the correct position, the CPS tells the DME to fire the spark. The DME uses computer precision to fire the spark.

The CPS can get dirty or contaminated or it can fail altogether. Since the CPS is telling the DME when to fire the spark its easy to see that a faulty CPS can cause rough idle, misfire, loss of power, backfire and if it fails completely then the DME doesn't know what position the piston is in and the car will crank but it won't start.

Cam Position Sensor - (CMP) - The cam sensor (I'll use CMP) is mounted at the cam gear or further back in the head. It measures the position of the cam and is used to control when fuel is injected. Each piston comes to the top of the cylinder twice in each rotation of the engine. During the ignition stroke all of the valves are closed and the air is compressed. Fuel is injected and the plug fires. The ignition stroke only happens every other stroke so the cam sensor is used to tell the DME which stroke the engine is in and when its time to open the fuel injectors and fire the spark. Since it measures the position of the cams its easy to see that a faulty cam sensor can cause extended cranking to start, rough idle, idle bounce, back fire, loss of power, misfires.

The CMP works together with the CPS to tell the DME when the piston is in the correct postion and the correct stroke to inject fuel and spark so essentially they both control spark and fuel injection.

MAF - The MAF is located between the air filter and the throttle body. Its usually a large box with a cable going to it. The MAF uses a delicate sensor inside to measure the volume of air passing through it. This info is sent to the DME and the DME adjusts the amount of fuel injected into the cylinder for the correct fuel/air ratio. Since the MAF is on the vacuum side of the engine it's easy to see that all vacuum hoses and connections need to be tight for it to work propely. If air enters through a leak in a hose or connection then its entering after the MAF and its not being measured. The DME only measures what passes through the MAF. If 10% of the air is getting into the engine after the MAF then you'll get 10% less fuel and the engine will run lean.

A faulty MAF can cause rough idle, poor performance, poor gas mileage, idle bounce.

The MAF is easy to test. If your engine is idling/running rough, unplug the electrical connector for the MAF. This will cause the DME to use default values like it does in Open Loop. If the engine smooths out after a second or two then the MAF has a problem. If it stumbles and tries to stall then the MAF is probably ok. That sensor inside can get coated with dirt and oil (especially if you use a filter that requires oil). Remove the MAF and use MAF cleaner only. It cleans instantly, evaporates very quickly and doesn't leave a film. soaking it in rubbing alcohol works but takes longer. Reinstall the MAF and pug it in. If the idle doesn't improve then clean it again and reinstall. If cleaning it twice doesn't improve things then its probably time to replace the MAF.

Idle Air Control Valve (IACV or ICV) - The ICV opens and closes while the car is at idle and adjusts the idle speed. It consists of a motor and a vane/door that opens and closes a small window. A tube runs from the intake elbow to the ICV and another from the ICV to the intake manifold. Note that all air used by the ICV is metered air so air fuel mix is maintained. The ICV allows air to bypass the throttle body. It acts just like an auto choke in older cars. Air comes into it from the intake elbow and passes through a small window then out to the intake manifold. That window has a metal vane that is moved by the motor to open and close the window just enough to maintain a steady idle.

The vane and whole ICV assembley can get dirty which causes the vane to stick. Removing it is a bit of a pain since its located under the intake manifold. Spray it with carb cleaner until its completely clean. Inspect the little vane for scratches that could cause it to malfunction. Once its clean you should be able to spin the body back and forth and see the vane move freely. If its cleaned properly it will take almost no effort to get the vane to move.
A dirty/faulty ICV will cause rough idle, mild to sever idle bounce and if it gets stuck completely open it can cause the car to stall, misfire, backfire at all RPM's

Intake Air Temp Sensor (IAT) - Very simple sensor. Its installed in the intake manifold and measures the temp of the air passing through the manifold. Cold air is dense, warm air is less dense. Denser air has more air molecules so it needs more fuel for the proper fuel air ratio.
Failure of this will tell the DME to add more or less fuel (rich/lean) depending on how it failed. Loss of power, poor fuel consumption, poor idle are all possible symptoms.

O2 Sensor - Pre cat - 92-95 M50's and M42's have only one O2 sensor and its located just before the cats. 96-98 M52's have two precat sensors and two post cat sensors, 96-98 318's and 323's have one precat and one post cat sensor.

O2 sensors are used to measure all the exhaust gases. They are made up of a sensor and a heater. They need to be at the correct operating temp to work properly. This operating temp has very little to do with the temp of the cats. The precat sensor heats up and tells the DME its ready to switch to Closed Loop. Until it heats up the engine will operate in Open Loop even if all the other sensors say its ok to switch to Closed Loop. This is why a bad precat sensor causes such poor gas mileage and poor overal preformance. The internal sensor can fail or the heater can fail. In either case the entire sensor needs to be replaced. On M52's there is one precat sensor for cylinders 1-3 (bank 1) and one for 4-6 (bank 2). All other e36's have one precat sensor. Under normal operation it sends the DME specific voltage or resistance readings. These have to be within a certain range. The DME will adjust fuel air mix acording to the readings to get them to their optimal range. If the readings are at the top or bottom of the range for too long and the adjustments the DME makes have no effect then you get a Check Engine Light and a Error Code.

O2 Sensor _ Post Cat - As disscussed above..92-95 cars don't have a post cat sensor. 96-98 cars will have one post cat sensor for each cat. The post cat sensors measure the gases comming out of the cat. Nothing changes because of there readings. It is only a sensor. if the readings are in the wrong range for too long you will get a Check Engine light and an Error Code. This doesn't mean the cats are bad or the sensor is bad. It simply means the readings were out of acceptable range for too long. If the car is running too rich or too lean because of another issue its very possible to get a post cat error code. Chances are if you get a post cat code you'll have more than one code indicating multiple problems. Just like the pre cat sensors, the post cat sensors need to heat up. Faulty Post Cat sensors generally won't affect performance. They only measure the output of the cats.

Coolant Temp Sensor - This is located in the head and sends a signal to the DME that the engine is up to correct operating temp so it can switch from Open Loop to Closed Loop. Failure of this sensor can cause the car to remain in Open Loop.

Throttle Position Sensor (TPS) - this is located on the throttle body. On cars with ASC there will be one located on the Secondary Throttle Body in addition to the Primary Throttle Body. These are a simple vairable resistor. The DME compares the voltage going in to the voltage comming out so it knows exactly what position the throttle is in. The DME measures changes in this reading to make the RPM's drop or increase. There are specs available for the correct Ohm readings for these. A faulty TPS will cause high/low engine rev, poor performance and generally anything you might associate with things like a loose or faulty throttle cable.

Here is the test procedure for the TPS

Harness connector disconnected, ignition on:
Terminals - 3 and ground in harness connector = 5 VDC
Harness connector disconnected, ignition off:
Terminals - 1 and 3 at sensor terminals = 4k ohms
Throttle plate rotated from idle to full throttle position:
Terminals - 1 and 2 at sensor terminals = Variable from1-4k ohms without interruption

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