The Lambda Sensor is responsible for monitoring the oxygen content in the exhaust gasses from the engine.
It is used mainly for 2 functions. The first is for fuel economy and the correct running of the engine. The second is to ensure the catalytic converter is operating correctly.There is little point in an engine burning more fuel than it need to. A Rich fuel mixture is very damaging to the Catalytic converter and a lean mixture will cause the engine to overheat and may cause valve recession.

2 of the most common types of Lambda sensors are Zirconia and Titania type sensors.

The Zirconia type Lambda sensor compares the oxygen content of the atmosphere with the oxygen content of the exhaust gasses. It has a small opening above where it is fitted into the exhaust or manifold. The larger the difference in the oxygen readings the larger the signal voltage. The sensor can be a  0 to1 volt or inverted as a  1 to 0 volt. The voltage will fluctuate between 0.2 and 0.8 volts when operating correctly.
In a  0-1 volt sensor 0.2 volts would be a lean mixture and 0.8 volts would be a rich mixture.
 

It is also supplied with a 5 volt reference signal.

Common terms used when talking of lambda sensors are "Lambda1" "Closed Loop" and "Open Loop"
Lambda 1 refers to the stoichiometric air / Fuel mixture which is 14.7 to 1 - The perfect Air / Fuel Mixture
When the Exhaust gasses are measuring lambda 1 the engine is running at optimum fueling conditions.

Open loop refers to the state of the lambda control circuit. Open loop occurs when an engine is warming up. The rich mixture that the engine needs is ignored until it reaches operating temperature. When the Lambda sensor reaches operating temperature the system switches to closed loop. The circuit can also go Open loop on harsh driving conditions ie full throttle.

Closed loop refers to the system running correctly within its acceptable limits. Both the Catalytic converter and Lambda sensor are at their correct operating temperatures and are working within their acceptable limits.

The operating temperature of both sensors are controlled by the use of a heater circuit integrated into the body of the sensor. This ensures the engine can reach optimum running conditions quicker.

Most single bank engines will have 2 lambda sensors. One either side of the catalytic converter. The front one is used to control the fuel mixture and the rear one is used to ensure the front lambda sensor and catalytic converter are operating correctly.

Testing with a Voltmeter: The Heater circuit is supplied with battery voltage.The heater control wire will switch depending on the temperature control. The heater wires are often white in colour.
The Signal wire will fluctuate between 0.2 and 0.8 volts. If an average reading is taken it would measure 4 - 4.5 volts for a correctly working system. The signal wire is often Black. The earth cable which may be earthed and monitored through the engine control unit is often Gray.

Testing with an Oscilloscope:

Zirconia Lambda Sensor: The signal wire oscillations caused by  the fuel control system switching rich and lean can be seen.
Excessive hashing can indicate Ignition miss fires.
 

Titania Lambda Sensor: The signal wire oscillations are much smoother as the resistance changes in the circuit.

Reading taken with a Pico Oscilloscope.
 

Testing this way ensures you can diagnose a problem correctly. Diagnostic scanners can show a lambda fault which is often nothing to do with the sensor being faulty. It can be a fault with the ignition system or an air leak that is causing the Engine control unit to produce an incorrect mixture. As soon as the Lambda sensor steps outside of its acceptable limits the engine management light will come on.