Ok. From a very basic standpoint the power phases are like the firing order of the cyclinders in your car.
For instance, if the board (or graphics card) has eight phase power, then the incoming voltage from the PSU is made available to eight buck converters* (each being a transistor to take the voltage load from the PSU, and 2 switches that alternately charge and discharge the load from an inductor) - so these eight converters fire through one cycle (i.e. at 45° if one cycle equals 360°) through their seperate switches, and are rejoined into a single board input. Technically, dividing the incoming current through seperate switches/phases reduces heat and voltage drops... So, the more converters (or phases) then, theoretically, the cleaner the input power since you have incoming power more times per cycle -less opportunity for current drop between discharges.
Using the car spark plug analogy, it would be akin to an 8-phase board being a V-8 engine, while a 4-phase board would be an in-line/V-4 .
The +2 part of the equation (8+2, 12+2 etc.) refers to the same principle, but in this instance the current is going to the memory controller (either chipset, or in the CPU if the controller is part of the CPU package).
* this converts the load from the PSU into a load that the CPU and board can utilise (a DC-to-DC converter).
