Classic Mode is usually the default mode of game play of Fire Emblem. When a character's HP is lowered to 0, the character will permanently die and cannot be used in any chapter after their death. Players will need to restart the chapter (or start from the last save spot in Fire Emblem: Shadow Dragon if they want to get them back normally. An Aum Staff or Valkyrie Staff can be used to revive a character in the games in which these items exists. However there are usually only 1 that exists in those games.
Certain characters in the game such as Seth in Fire Emblem: The Sacred Stones or Frederick in Awakening will not necessarily "die" in this mode due to their importance in the story. However, they will be unable to be used due to their injuries.