A vehicle engine requires oil to be pumped around it for lubrication of parts - to prevent wear - and for cooling. The oil collects in the sump of the engine and is pumped at pressure to the top of the engine and to various important components. From there it drips back down through the engine and returns to the sump.
The pressure coming out of the pump should be kept high enough to ensure that a suitable amount of oil is provided, however too much pressure would cause leaks, as the oil seeps out between the gaskets between engine components. It is particularly important when the oil is cold, and therefore more viscous.
The oil pressure relief valve is often as simple as a small ball bearing on a spring. It opens at the optimum pressure, and dumps any surplus oil straight back into the sump.
The valve is usually convenient to access, being held in by a hexagonal nut on the engine block. For design simplicity it is often found near the feed pipe where it enters the block from the oil pump. Your auto maintenance manual should show its exact location for your vehicle.
The valve can fail in a couple of ways. Firstly, it can jam shut - thereby increasing the pressure of oil in the engine. Symptoms will show up as an unusual amount of oil leaking from the engine - and a big dark patch on your driveway! If you are lucky enough to have an oil pressure gauge, you will see the needle rise. Unfortunately, if not caught early enough, the pressure may rise high enough to break the gauge, spilling oil everywhere and resulting in James Bond style billowing smoke as the oil lands on hot components.
If the valve sticks open, there will be a sudden drop in pressure. Hopefully, the oil light should come on, and at this point, you should stop the engine as soon as possible! Don't crash, but your engine really will be ruined beyond repair within the space of a minute if it's allowed to continue running.
Very occasionally, parts can break off the valve and either flow into the oil (stuck open) or block a passageway (stuck closed).
Repair can be as simple as removing the valve and refitting it. Often, the disturbance is enough to fix it. If the spring or bearing are suspected, then new parts should be sourced - they will not be expensive.
A common modification to a performance engine is to uprate the spring so that a higher pressure is maintained and the oil flows quicker - this is often combined with an oil cooler.
Thanks to shro0m for providing a vivid image of the oil pressure gauge's breaking, and pointing out just how important it is to stop when the oil pressure drops. Also, to Kidas for pointing out that people might want to know how to locate it.