I don't think "hardcore" gamer means anything, really... It's just an adjective used to say that a person is really into gaming, I suppose.
On the other hand, I've heard the term "Core" Gamer to be a term that differentiates a type of gamer from "Casual" Gamer.
A Core Gamer is a demographic type that is enthusiastic about gaming and is likely to be interested in not just playing games, but delving into the mechanics of games, modding, design... participates in social gaming communities, etc.
The industry might see a Core Gamer as one who prefers the deeper and more complex gameplay of PC and/or console titles such as Call Of Duty, WoW, Modern Warfare, Red Dead Redeption, Starcraft, Civilization 5, etc... you know, AAA titles. I tend to think a Core Gamer is as likely to play Bejeweled or Zuma or even Farmville as well, but that gaming for them is a serious hobby. In fact, many probably like tabletop gaming as well.
I also think that just about every single person on this site is a Core Gamer. This community's interest in Baldur's Gate, XIII, Fallout 1 and 2 and the like is as strong as their (our) interest in Dragon Age 2 or Duke Nukem Forever.
The Casual Gamer might only enjoy some occasional gaming and might focus on titles with a social focus, like Rock Band or Wii Sports or several of the Facebook sort of games.
I think it is really a matter of interest. A Core Gamer will devour any game they think is fun and challenging, and probably spends most of their free time gaming whereas the casual gamer is far less obsessive, isn't hellbent on finishing every title, and doesn't salivate about new titles on release day.
I hear the terms used as derision, as if to say casual gamers (and thus casual games) are inferior to Core Gamers (and thus Core Games), but I don't that's either fair or right.
I have traditionally thought of myself as a Core Gamer, though my behavior of late is more casual because I'm focused on school right now. I tend to ramp up in the summers and get really obsessive.
Anyway... that's my 2 cents on this.