Various people have tried. Neil Gaiman called political correctness, "treating other people with respect," and certainly there's an element of that there. When someone says, "This isn't politically correct," it generally does mean, "This doesn't treat other people with respect," but the thing is that what is politically correct doesn't necessarily treat other people with respect either.
Likewise, what is politically correct changes over time in ways that don't necessarily match the ways treating other people with respect change.
So what does it really mean?
Strangely, I think it was best summed up in a statement, which was made before the modern idea of political correctness solidified and never mentioned the idea of "politically correct", from Lee Atwater.
In an interview in 1981 someone asked about the way Reagan's policies of cutting aid gain him the support of formerly democratic racist voters of the '68 Wallace bloc. You don't need to understand what any of the previous sentence means to understand the answer.
Do be aware, though, that the answer includes a racial slur repeated four times in rapid succession.
You start out in 1954 by saying, "Nigger, nigger, nigger." By 1968 you can't say "nigger" — that hurts you. Backfires. So you say stuff like forced busing, states' rights and all that stuff.
The message didn't really change, if I'd gone on for a couple more sentences we'd see Atwater water things down with "subconsciously maybe" and so forth, but the core is right there, and it's basically a perfect show of what political correctness really is.
The message hasn't changed. The words have. Suddenly (well, gradually) the racial slur is something you don't say in polite company and it makes you look like an asshole. So, what's an asshole to do? Use different words to say the same thing. (If you can.)
That's the core of it.
At some point in the progression of any group's rights it becomes uncouth to verbally treat them like shit in public. At that point the politically correct thing is to not treat them like shit in public.
This is, in fact, a necessary component of treating other people with respect, but it is not sufficient.
The result is that people view political correctness in very different ways.
Some people don't want to be assholes. As such, it doesn't even occur to them to want to say the 1954 thing. Being politically correct isn't bothersome to them, it's not even a concern. If they learn they've been doing something that isn't politically correct they'll stop doing it (or try, depending on how ingrained it is) because they don't want to be an asshole. The fact that it's politically incorrect won't even play a role in them stopping.
Some people don't really mind being assholes, but they certainly don't want to be seen as being assholes. To them, then, political correctness is a way to be the assholes they want to be, or appeal to assholes whose support they want, without saying, "I'm an asshole," directly. For them, then, being politically correct is a tool to do something without overt, or at least without obvious, indicators that they're doing it. In their view political correctness is sufficient.
And some people see the whole idea of not being an overt asshole as infringing on their right to say whatever the fuck they want. It isn't, because you can still say politically incorrect things (again, modern parlance), but there is a structural disincentive. They don't like the idea that they'll be branded as an asshole for refusing to hide the fact they're an asshole.
Political correctness is bad for them because once upon a time people had the privilege of spouting racial slurs (or whatever the assholicness in question is) without any push-back but now if they do that they're looked down on.
Others completely miss what the whole exercise is about and think that political correctness is a bunch of meaningless hoops to jump through. For them it seems like the important thing is that they know what is and what is not politically correct, rather than that they actually consider the content of their words or actions. As such saying, "I know this isn't politically correct, but..." is the same as tacking, "bless [possessive pronoun] heart," on to the end of a string of insults: it makes it all ok.*
There are various other views as well, but I'm running out of steam for this post so I'll wrap up.
Rule one is not enforced. In many ways it can't be enforced. There are too many ways to be an asshole and, moreover, making someone stop being an asshole requires changing their mind. Rules can't really do that. In other ways it shouldn't be enforced. Who gets to decide what being an asshole means if we make it against the rules to be an asshole?
But prohibitions on certain behaviors can be enforced --some legally, others informally.
Political correctness is an informal enforcement of the idea that you shouldn't be an overt bigoted asshole against groups we as a culture find worthy. It is, as a result, a very low bar to meet. For political correctness to be against you you have to be behind the culture as a whole. You've got to be trying to use the 1954 language in 1968 or, worse still, 1980 (the interview was in 1981, recall.) By 2015 most people wouldn't even think to call the 1954 language "politically incorrect"; it's just wrong in the eyes (ears?) of the dominant culture.
The way political correctness is enforced isn't any goose stepping jackbooted PC police, it's instead by people hearing the politically incorrect thing and thinking, "What an asshole," about the person saying it.
That's political correctness. Being an overt asshole toward a group that is no longer seen as an acceptable target is politically incorrect and the punishment for such overt assholistry is that the politically incorrect words or actions will be seen as assholic.
What is considered overt and who no longer constitutes an acceptable target changes over time.
And that, basically, is the whole thing.
Or, at least, that's my attempt to describe the modern meaning of political correctness.
* He's a lying, thieving, cheating, scum-sucking, dog-licking, spit-swimming, spider-eating, goat-hugging, dung-smearing, pig-kissing, frog-swallowing, mud-biting, cow-tipping, toilet-swabbing, cud-shoeing, window-washing, half-warped, apple-polishing, worm-witted, chicken-hearted, lamb-lusting, nefarious, untrustworthy, nasty person, bless his heart.