Powerplay was kind of "tacked onto the end of the game" back in version 1.3. Prior to that, all we had were the minor factions and the superpowers. People said they wanted to formally "join" or "pledge" to a superpower or a minor faction and form EVE-style clans and player groups around their choices; ED said no, but Powerplay was implemented for these people instead as a substitute.
The Powerplay powers operate on a separate plane to the superpowers and minor factions; they have a superpower they are affiliated with, but this affiliation does not directly control the "allegiance" of a system, nor do they directly alter the minor factions within a system. However, there is some interaction between the Powers and the other levels of government.
If two Powers are both affiliated to the same superpower, they are considered to be on the "same side", in the sense that violent attacks causing death are frowned upon, rather than rewarded. In other words, if Aisling Duval supporters attack Denton Petraeus ships and destroy them, they don't get merits or doing so, whereas they will get merits if they destroy ships of non-Imperial Powers. They can still work against each other (eg. by acts of piracy stealing their rival's powerplay materials), they're just not supposed to kill each other.
Powers do not directly alter the minor factions that are in charge of a system, but there is a consideration that makes Powerplayers want to change the factions that are in charge of systems. Each Power has "strengths" and "weaknesses", which mean that certain faction types are cheaper or more expensive for them to control. So, to increase their profitability, Powerplayers do try to flip systems over to governments which their Power can more easily control. Ironically, this can have the effect of Powerplayers wanting to flip systems away from the Allegiance that their Power is nominally aligned to.
For example: it's my understanding that Aisling Duval is actually weak against Dictatorships and Patronages, the two government types that are most commonly encountered in the Empire. So Aisling players tend to want to try to flip systems under their control away from Empire-aligned to Independent or even Federal-aligned factions.
Three of the Power leaders - namely, Arissa Lavigny-Duval, Zach Hudson and Ed Mahon, also happen to be the heads of state of the Superpowers they are aligned with. This does not directly influence the Superpower in question, or even mean that those particular powers are better or more favoured by the Superpower in any way. As stated above, Powerplay players aligned to one of these Powers can even directly oppose the interests of that Superpower. Another example: I believe Mahon favours Corporate factions, and the mechanism of Powerplay means he does not really care whether those Corporate factions are Alliance, Federation, Empire or Independent. Faced with a choice between an Alliance democracy and an Independent Corporate minor faction, Mahon powerplayers will choose to back the Indie Corporate.