A chain of around sixty six of these communications satellites have been in orbit for some years, the name deriving from the atomic number of the metallic element Iridium - it was originally intended to launch 77 of them.
The company that operates the Iridium satellites has begun 'de-orbiting' and replacing them with a new version that will no longer produce the beautiful and much-enjoyed flare. If you've never seen one, you have around a year to do so: you can find predictions online at the free site Heavens Above (You have to enter your location for a prediction accurate to the second!)
Photographing the flares is straightforward: put your DSLR on a tripod with the lens set on infinity and settings of 1600 ISO and f5.6. Point the camera towards the part of the sky where a flare is expected: the satellite will initially be visible as a bright starlike object. At the predicted time (with exposure on 'bulb') take a series of 10 - 15 second exposures.
In best Blue Peter style, here's one I made earlier: last night in fact. It occurred near the constellation Taurus, close to the Pleiades ('Seven Sisters') but wasn't particularly bright... The other image is of Orion, everyone's favourite star group.