There are many awesome pictures of this phenomenon such as this one:
What is the physical explanation of how this occurs? Does it occur only when the airplane crosses the sound barrier or does it continue throughout the whole supersonic regime if enough water vapour is present in the atmosphere? What factors does this condensation depend on?
It is important to note that the vehicle is not traveling at supersonic speed when condensation collars (rockets) or condensation cones (aircraft) occur. It is merely close to the speed of sound.
Air is accelerated by the displacement effect of the vehicle - it pushes the air aside and around itself, causing the flow to accelerate and to become supersonic relative to the vehicle. This acceleration is directly connected with lower pressure (see this answer for a detailed explanation), and air with high relative humidity will be pushed beyond its dew point, causing the water vapor to condensate. Please note that the onset of condensation is slightly delayed, so the fog is not strictly indicating the region of lowest pressure. Also, the forward cone is not the same as the forward shock front of a lambda shock.
However, the rear boundary of the condensation is indeed caused by a shockwave. The low pressure area is instantly ended by a compression shock, when the supersonic flow around the aircraft is abruptly decelerated to subsonic speed by a compression shock. This raises the local air pressure back to around ambient pressure, so the capacity to absorb water is raised again and the fog evaporates. Again, this happens with a slight delay, so the end of the condensation area does not coincide with the compression shock but follows it.
Since the low pressure area is caused by the aircraft, it travels with the aircraft, and so does the condensation effect. The air around the aircraft or rocket is accelerated towards the vehicle by its pressure field, which causes the condensation. Now the air is so fast relative to the vehicle that it is decelerated and recompressed by a straight shock, causing the condensation to disappear again.
A relatively blunt vehicle traveling at close to Mach 1 and humid, almost saturated air are the factors which help this effect to occur. Note that you can witness similar condensation effects over the wings of airliners in humid air during the landing phase, when flaps are fully deployed. Here, however, the condensation does not end abruptly, since no shock is involved. Also, the disturbances produced by the aircraft do not reach out so far at lower speed, so the condensation happens only close to the aircraft's surface. Bluntness helps: The cockpit causes its own small condensation area shortly past its maximum thickness, where suction due to displacement is highest.
