Another classic for the FAQ!
A blowoff valve (also known as a BOV or compressor relief valve) releases the pressure surge in the intake system between the turbocharger and the throttle body when the turbocharger is producing boost and the throttle is suddenly closed. That's why a BOV makes noise during a shift. It's triggered by the vacuum signal created in the intake manifold when the throttle closes between gears.
There are two main reasons BOV's exist. The first is to relieve the backpressure so that the turbocharger doesn't suddenly slow down between shifts, thus reducing lag between gears. The second is because that sudden surge when the throttle is closed can be very damaging to the turbocharger, especially if the intake happens to be the "right" length to reflect the pulse in such a way as to reinforce it.
There are two main kinds of BOV's, as well: Vented, and recirculating. Vented (or "atmospheric") BOV's dump the excess pressure to the outside world, while recirculating ones redirect it back into the intake side of the compressor. Every factory turbo vehicle that I can think of equipped with a BOV has a recirculating type. Cars with MAF-type engine control systems require it, because otherwise air that has been metered never makes it to the engine and it runs very rich every time the BOV cycles. Plus, not everyone thinks the sound of an atmospheric BOV is cool... Personally, I think it announces, "Hey, look at me throwing boost away! I'd rather make noise than power!" :p