Casual and Classic modes in the game mode settings in Awakening
Classic Mode is, in most titles, the only 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 map save or save spot in Fire Emblem: Radiant Dawn and Fire Emblem: Shadow Dragon) if the player wants to get a unit back within game mechanics, An Aum Staff or Valkyrie Staff can be used to revive a character. However these staves are unique, very limited usage, and only even appear in Ankoku Ryū to Hikari no Tsurugi and Genealogy of the Holy War.
It should be noted that certain characters in the series, such as Seth in Fire Emblem: The Sacred Stones or Frederick in Fire Emblem: Awakening, will not die in a narrative sense, they continue to appear in the story afterwards. They will be unable to actually be used in gameplay due to their injuries.
Starting in Shin Monshō no Nazo ~Hikari to Kage no Eiyū~ players can opt-out of this mode and instead play Casual Mode, in which characters do not die and return the next map.