A Hardcore character is not different from a normal character except one very interesting and major point. When a Hardcore character dies, the game ends, and you cannot simply reload your last save point because the character is gone forever. In effect, the game treats you like a mortal human being rather than the almost immortal non-Hardcore characters, accordingly called "Softcore" by Hardcore players.
Hardcore characters are displayed in a more aggressive pose in the Character Selection Screen, depicting their tense state of play. When a Hardcore character is killed, they are not removed from the Character Selection Screen, remaining there as a robed figure reminiscient of the Grim Reaper, reminding players of their loss until the character is deleted by the player. Other than these, there is no other difference in gameplay terms. Dead hardcore characters will even retain whatever position on the Hardcore Ladder they may have achieved (albeit with their names in gray) until they have been surpassed by other players enough to knock them off. As such, Hardcore mode is regarded by many as nothing more than a more realistic playing experience by the community.
However, there is a simple, perfectly legitimate, but often frowned upon workaround to the permanent death of the player's non-multiplayer Hardcore characters. The player can simply copy the character's file in the game directory, paste them elsewhere, and re-paste them back to the directory in order to resurrect the character. This cannot be done on Battle.net realms because the character's data is stored in the server itself.