First off, I play games. A lot.
I've been playing games for as long as I can remember. Atari, Super Nintendo, Sega Genesis, Nintendo 64, PlayStation 2, Xbox 360, PlayStation 3, PC, these have all existed in my household at one time or another. This isn't even mentioning the huge amount of PlayStation, Xbox, Gamecube, and Wii titles that I've played at friends houses. Gaming has always been a hobby of mine.
But, in my own opinion, there is no such thing as a 'hardcore' gamer. Generally speaking, whatever else people might say alongside "I am a hardcore gamer," the crux of the statement basically comes down to 'I don't just play games on my iPhone or on Facebook or on Flash-based websites," etc. 'Hardcore' gamer is a term that some people (and they're often quite aggressive) use to differentiate themselves from what they would call the 'casual' gamer.
You can think of them as along the same line as PC or Console elitists, only on a bit of a broader scale. Basically, they take gaming extremely seriously, and think that anyone who plays smaller (read: casual) games is a stain upon the community.
In another time, before 'casual' gaming became so widespread, they would likely have been referred to as the 'core' audience. You know, the audience (some people say) Nintendo abandoned when they released the Wii, or the audience that (some people say) Microsoft seems to be forgetting about as it tries to make the 360 a full multimedia centre. They would, at one point, have been the people who actively followed the release dates of the big upcoming titles, because they were acutely interested in perhaps purchasing them, rather than just being one of those people who would play a game if it were available, but not actively track which games are coming out when.
Basically, in the 80s, 90s, and (arguably) early 2000s, when video gaming was just picking up speed as a medium of interactive entertainment, today's 'hardcore' gamer would have just been referred to as a 'gamer.' Probably with a little bit of scorn.
But since there are so many smaller titles out now, and so many people who have consoles/PCs only to play a couple of games a year (this is where someone might spit the word 'CoD!' with a venomous hatred), the (often snobby) gamers who have a wide interest in video-gaming as a whole feel they need a word to separate themselves from those who view the medium in a narrower scope. This is your so-called 'hardcore' gamer.
I don't think the term should exist. I don't think the term 'casual gamer' should exist, either. The idea of hardcore vs. casual is like East vs. West: you're splitting into two categories what, in reality, is actually a much more diverse and multifaceted grouping of people.
TL;DR: it's an elitist term, at least in my view, because some people can't stand the idea of the 'casual' gamer, and feel the need to make a distinction for themselves.