Pressure relief valve is self explanatory, in a general sense, I think. You need a Pressure Relief Valve anywhere there is a need to relieve pressure.
The hot water tank in your home has a PRV. It is set up so that if the pressure inside the water tank gets too high, for any reason, it will open and vent the hot water and steam to atmosphere.
This usually means all over your floor! : -)
As it pertains to your compressed air system and your air compressor however, you will want to be sure that the receiver / air tank / reservoir into which your compressor assembly pumps air, has a working PRV.
When your air system calls for air, your compressor cuts in. It gulps in free air through its intake filter and scrunches that air into your receiver, continuing the process until the pressure inside the receiver reaches a high cut out pressure level. At this time, your compressor should stop.
Compressed air is explosive when it is released suddenly, and you sure do not want to be anywhere around if a compressed air receiver should disintegrate due to unsafe high pressure levels inside. It is a bomb!
To help ensure that this does not happen the compressed air receiver has a PRV either on the top or to one side or the other of the tank, or as part of the check valve / pressure switch / PRV assembly, where the air line from the compressor head enters the air tank.
Some of them look a bit like this.
There is a ring on the end of the PRV which, when pulled, will allow air from the receiver to vent to atmosphere.
Some PRV's rely on -sticktion-; that is the friction between its parts, to generate enough internal PRV resistance to keep the PRV closed.
Other PRV's have an internal spring performing the same task.
When the pressure inside the reservoir reaches the PRV set point, the air pressure acting on the piston inside the PRV overcomes the sticktion or spring pressure, forcing the piston out, and opening the flow path from the air line or tank to atmosphere, venting over pressure.
While I have never had a PRV on a receiver open due to over-pressure inside the tank, having the PRV there provides a critical and mandatory safety assurance.
For the same reason, you never want to have a source of pressure that can increase in any air system, without having a PRV installed to protect that air circuit.
It is cheap insurance, and having a working PRV in an air circuit could save a life.