The relationship between language and power is complex. The basis of the relationship is rhetorical, that is, if you speak the language expected of you in whatever rhetorical situation you find yourself in, you'll be more likely to be listened to and, hence, have power with your audience. There are, however, specific languages of power. English is a good example, and the form spoken by middle America, what you'd consider a lack of accent, is the standard for American English. For centuries, French was the language of power, and prior to French, it was Latin. For the Brits, it's a dialect of English--the Queen's English--taught at Oxford and Cambridge. Listen to a BBC news commentator for an example.
Given your writing and the fluency you show, I'm betting most of you grew up in middle class or upper class homes whose first language was American English. You have, in all likelihood, spoken and been exposed to the language of power in contemporary America (and most of the world) all your lives. This makes it more difficult to see just how tied your language is to power. Think, however, of situations in which you've seen somebody made fun of for their dialect. Country speak is a good example. Folks with an accent associated with rural life are less likely to be taken seriously than those who developed their language or mimic that of urban areas.
Other language cues to power are how you use evidence to support your opinions. As I said in the previous post, we associate professionalism and college training with specific formats of language use. Learning to use a specific jargon is another marker of being an insider in a discourse communities. It's one of many reasons it's a good idea to listen a lot and not speak when you enter into a new discourse community. You want to mark the formats, language use, and jargon of that community. For example, each of you will develop, if you already haven't, the jargon associated with your profession. Here think of geekdom and the language specific to it. Since it's true that passion and expertise in technological fields is associated with certain ideas, concepts, and words, geek speak is quickly assuming an association with power.
In short, the language of power is the language of those in control; and, language use is one of the many markers of who's in and who's out.