Eh, we're getting to the definitions stage of OT arguments again. Pedant to the rescue!
Hegemony is not a word in common usage, it's international politics jargon. Its meaning varies a bit according to analysts or scientists working definitions but the general meaning of it is quite well-established. It is a level of influence over other states that is much higher than ordinary but below direct control. It means other states listen when you speak and usually try to follow your advice. It means that states trying to better their own position either work with you or they look for ways to get around you.
It also means (on this I think
@IsadorBG is a bit vague) that within your sphere of influence you have no equal, no other state whose advice is taken just as seriously. The only way you can have two or more hegemons is if each has their sphere of influence. During the Cold War, SU and US were both hegemons within their own sphere, but they were never, and could never be, co-hegemonic over the same set of states. (There's a good argument that the US was a global hegemon because its sphere of influence was worldwide but the SU's more geographically limited sphere was not a part of it.)
The word hegemon is Greek and its first application was to the influence Athens exercised over its allies in the Delian League. In its early years this control was through informal influence, later on it mutated into more direct forms of control which are even characterized by many historians as an Athenian Empire. You don't need a word like hegemony for that later stage, it's useful only for the early stage.
As to the time period, the word has been applied to the US more often than to any other state in history. In Europe the US only exercised formal control over parts of Germany and Austria after WW2 for some time, it never applied military power to keep its allies in line, and yet its voice was always heard and usually followed, its stance was decisive for most alliance decisions. It makes no sense to describe NATO policies as the outcome of negotiations between equals. One can't accurately describe this without discussing how much weight the US could throw around.
The terms great power or superpower refer to something entirely different, namely the capacity to exercise military force on a scale that's matched only by others of that rank. A state can be a great power without ever having hegemony over any other state. Conversely, a regional hegemon need not be considered a great power on a global scale. Because one can exercise influence through other means than military power, e.g. through economic or cultural contacts, it's possible that a state that doesn't count as a military great power is nevertheless hegemonic. The influence of Germany within the EU is a neat example IMO.
Edit: fixed a misplaced comma.