Well, people who have picked up Halo 3 or whatever the PS3 equivalent is a couple of years back and have played it online like to think that they're at the heart of gaming. They're the ones who call themselves "hardcore", because they're self-important teenage boys who like to think that their opinion matters, and who like to solve their real-life bullying issues by bullying people on the internet (lol).
On the other side of things, there's the "casuals", who are people that are new to gaming Maybe they play Wii Sports or maybe they like Farmville - they're the new kids on the block who like to claim they're "gamers" because they've played Cooking Mama.
What about those people that resent these terms? The gamers that identify themselves as simply... gamers? Well, they're the ones that really enjoy games. They're the people that recognize that those who claim they are "core" are idiots, and "casuals" are people trying to get deeper into our hobby.
You have to see it from both sides, though. The casual gamers are simply trying to enter this great hobby of ours, while the "core" gamers are worried that this new torrent of new people are causing developers to dumb down the games they make to accommodate this new audience. Both sides have good arguments, but please for the love of god stop calling yourself a "hardcore" gamer. All it does is make you look stupid.