Well, I can try. Let's say you live in a suburb of Atlanta (notice the cap--it's a proper noun). You could also refer to Atlanta more formally as the City of Atlanta (caps again).
People in the suburbs might say they are going into the City, and the context implies that Atlanta is being referred to, albeit indirectly. But normally, the word "city" standing alone, would not refer to a particular, unique place, and you would not capitalize it.
You might say something like "Once we got off the backroads, we were driving through the city," but in this case you are not really referring to any PARTICULAR city, just "a city" (as opposed to the country). So no caps there.
See the distinction?
With respect to the "group" thing. If you are trying to refer to ALL democrats, then you are, effectively, referring to the Democratic Party. On the other hand, if you are just referring to some subset of democrats, that's not "the Democrats" (the Democratic Party). It's just a group of people who happen to be Democrats. They are a group of "democrats" (small "d"). It serves as more of an adjective than a noun in that case. Or, if it is a noun, it's a common noun, not a proper noun.
Once again, the wiki article I quoted said:
a common noun...refers to... non-unique instances of a specific class (a city, another planet, these persons, our corporation).
That's just the way I would interpret it. I could be wrong.