They basically constitute a little power supply on the motherboard, and they convert the 12V supplied by the (regular) power supply to the voltage required by the CPU (in the neighborhood of 1-1.5V, and can change dynamically). A quality power supply is important for the system as a whole, and the same goes again for the CPUs little dedicated power supply.
More power phases is a good thing because then the load on each power phase is reduced. That means they generate less heat, and that's great because heat makes them less efficient and stable. And in the worst case scenario, the VRM (Voltage Regulator Module, with X number of power phases) can even overheat and be damaged.
For more information about VRM, check
the excellent tutorial at Hardwaresecrets, or the
VRM/motherboard guide at overclockers.net.
PS: Haswell moved part of the VRM onto the CPU itself. This is part of the reason Haswell runs hotter than Ivy Bridge.