When a character's HP is lowered to 0, the character will die and cannot be used in any chapter after their death. When players lose units, this will not cause a Game Over. However, the death of a Lord, the player's Avatar, or an important ally that needs to be protected during a chapter will cause a Game Over if they are killed. This was the set up in all Fire Emblem games prior to Fire Emblem: Shin Monshō no Nazo ~Hikari to Kage no Eiyū~ where the permanent death feature could not be disabled.
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 recover a lost unit. An Aum Staff or Valkyrie Staff (appearing only in Ankoku Ryū to Hikari no Tsurugi and Genealogy of the Holy War) can be used to revive a character. However these staves are unique, and have very limited usage.
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 and continue to appear in the story afterwards, but will still be unable to participate in battle due to their injuries.
Shin Monshō no Nazo ~Hikari to Kage no Eiyū~ introduced Casual Mode which disables the permanent death feature.