Genres
FPS
Generically standing for "First Person Shooter", this is a very popular genre in the industry that loans itself quite readily to hardcore runs. With lots of enemies that all want you dead as a doornail, it takes a real badass to stay alive in a number of these games. A favorite of adrenaline junkies everywhere.
Recently, Dark Messiah of Might and Magic became one of the industry's first major "First Person Slashers", emphasizing the first person view and high, linear adventure while relying chiefly on melee combat instead of ranged. So one can't really call all of these games "Shooters" anymore.
RPG
"Role Playing Games", as they're called, were around a long time before video games. However, what they commonly amounted to was a lot of manual dicerolling that people didn't have a lot of time for. It was only natural to put these games behind a computer and let the machine do all the dicerolling for the player. Since then, RPGs have become some of the favorite past-times of hardcore gamers because of the incredible amount of control they typically give you over your characters. Unlike FPS games, where a player is given a gun and told to run with it, RPGs allow the player to finely craft their own gun with which to blast through the game.
Strategy
Strategy games, the crown of armchair generals everywhere, have long been a favorite for playing with finesse and skill. These games involve moving many units of various skills around to conquer unique obstacles. These are not as easy a fit into the rules of hardcore as some other games, since it is inevitable that some soldiers will die in all wars, but almost all games have some sort of mechanic that lends itself nicely to hardcore.