A person that spends most of their leisure time playing video games, or doing video-game-related activities (like discussing video games on forums, for instance).
You will note in this definition that in no way do I mention genre. The soccer mom who spends every spare moment glued to FarmVille, discussing how to get the next level and figuring out how to min-max coin production to get a shiny new upgraded farmhouse or what have you, is on equal level with the kid who can tell you all his favourite Pokemon strategies and how to beat every gym from the past three generations and the nuances of the personality of Cobalion as seen in the Swords of Justice movie, and on equal level with the person who is part of a DOTA team that lets everyone else know they're about to be wrecked by synching their avatars and player names to be that one guy from that one Old Spice commercial and can speak fluent jargon on level with the best WoW GM (gg no re) and has probably also learned how to say 'STOP FEEDING' in at least 5 languages, and is on equal level with the person who has spent more hours in gun fights in the latest Call of Gears of War Runny Gunny Fun than somebody who has been on two tours to Afghanistan and has picked up enough military speak and such a clear commanding voice telling other people what to do strategically that people assume them to be of high rank in the military when the closest they've done irl is that one time they went deer hunting last fall, and is on equal level with someone who keeps a stable of old computers alongside the newest DOS-boxes on their modern machine so they will always be assured of being able to play their favourite point-and-click and who says that at heart the experience isn't the same if you don't have to change out a floppy mid-game, and the -- okay, actually, I think you guys probably get the idea.
"Hardcore gamer" is often a title that people use not like a medal of honor but like a baseball bat. Specifically, it's a great way to get all No-True-Scotsman about who is and isn't part of the community. Much like how innovations in research come from people reaching out of their insular college departments to make connections with something entirely different, every type of gamer has a slightly different point of view and comes from a different background and will have unique insights to bring to the table. By sticking together and recognizing that one form of gaming is not more noble and above the rest, we'll have better discussions, better communities, and even better games as developers have bigger pools to look at and go "hm, how about we do an idea kind of like that, but modified?" for their new concepts.
That's the goal, anyway. Unfortunately it just gets co-opted by snobbery a lot of the time. Especially towards women, which is pretty discouraging. That's something we, as a community, need to look out for: how is the label of 'hardcore gamer' being used? Does it include women? gay people? people who aren't white? other marginalized groups? Because if a hardcore gamer is always a straight white dude, there's something hinky going on, and the term isn't being used to bring people together over a common interest, but instead as a fencepost to keep people out of the discussion.
tl;dr: do games make up a significant part of stuff you enjoy and talk about? grats ur a hardcore gamer.