Define the following.

[Eomund]

A quick media definition
Hardcore gamer: A person who lives to play video games.
Hardcore game: A game made for said Hardcore gamer.

Now I know that some people are "hardcore" when they don't meet the above requirements. The above requirements are more for mass consumption and are portrayed negatively because there is a small bias against such "fanatical game players."

In any case a "Hardcore Gamer" is someone who takes great joy and even pride in his gaming accomplishments and/or abilities. He/she can spend lots of time playing any certain game and not get bored. A person that takes a gaming console to a party or gathering to play with a group of people could be a hardcore gamer as well.
A "Hardcore Game" is a game that could require lots of attention to detail and needs lots of micromanaging. Most RPGs could fall into this category. Some FPS will fall into this category. A hardcore game must have depth.

I hope that works for everyone.