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Hardcore is generally a term for MMORPG players who play a lot (30+ hours a week), prefer challenging play (PvP or cutting edge PvE), and don't like rulesets that make the game easier. Often used in contrast to the term casual.
Blizzard has made more effort than many companies to make World of Warcraft friendly to the casual player, but definitely has lots of content for group-oriented, hardcore players with their focus on large raid dungeons (20-, 25-, and 40-man) and battleground PvP for more solo-oriented hardcore players.
It is unrelated to hardcore mode in the Diablo franchise, where there is a game mode wherein if a player character dies, his character is automatically deleted. World of Warcraft has no such mode - it's not really designed for that sort of gameplay, though people have wondered what such a mode would be like.