
Everybody who has ever tinkered with the exhaust system of a fuel injected car, has probably seen it sticking out of the exhaust pipe.
The lambda probe or sensor is located somewhere in the exhaust, before the catalitic converter.
Although more modern cars can have multiple Lambda probes before and after the catalitic converter. This way the ECU can monitor the difference between the exhaust gasses before and after they went through the catalitic converter. If there is no difference, this would mean that the cat could be broken.
But for Toyota's from the late '80's, and early '90's usually have only one probe that is mounted somewhere in the exhaustmanifold or the downpipe, leading to the cat.

A lambda probe from a Toyota Corolla
The Lambda probe is a important part in creating the right air/fuel mixture.
The Lambda probe measures how much air is still present in the exhaust gasses.
See the picture below for refrence. I've pointed out most of the components that make up a Lambda probe. The Lambda probe is hollow and air can get in to the air side of the probe through the small hole that creates an air opening.
The outside of this hollow part is encased in a ceramic layer from sirconiumdioxide.
All of this extends into the exhaust gas side of the probe. On this side, exhaust gasses can get into the probe through several small holes in the protecting cage.
Because there will be a difference in oxygen levels on the air and exhaustgas sides, the probe will be able to generate a small electric current. Often in the range of 0,1 and 0,8 Volts.
At 0,1 to 0,2 Volts, there was a lean mixture in the combustion chambers
(Not much difference in Oxygen levels).
At 0,6 tot 0,8 Volts, there was a rich mixture in the combustion chambers
(Big difference in Oxygen levels).
A Lambda probe will work properly when it has reached its operating temperature of somwhere around 200°C. It will reach this temperature on its own in the hot exhaust gasses, but when the engine has just made a cold start, it takes a while for everything to heat up and the ECU can't use the readings of the Lambda probe. The engine can run without them, but it has no idea if the mixture is right. This can lead to higher emissions of harmfull gasses. To counter this, some vehicles are equipped with a heating element in the lambda probe that will speed up the process. These probes can be recognised by the two additional wires for the heating element leading to the probe.

A cut through view of a Lamda sensor
With the signals comming from the Lambda probe, the ECU will try to adjust the air/fuel mixture by changing the amount of fuel injected into the cylinders.
This is called a 'closed-loop' system.
When the Lambda probe malfunctions and the ECU doesn't get any signal from the probe, there is a 'open-loop' situation. This could cause the engine to run bad or that the catalitic converter can't function properly, because the catalitic converter can only work with a correct mixture.
It could even be dangerous to the engine or other components, because the ECU can't see if the mixture is rich or lean and can't be corrected anymore.
Unfortunatly, you can't check if the lambda probe works correctly with a household Multimeter. The signal of the probe is already very small and it fluctuates rapidly because the oxygen levels change all the time. This makes it very hard to get a correct reading. The right tool for this would be a osilloscoop.