Classic Mode refers to the traditional gameplay mode used in the Fire Emblem since since Fire Emblem: Shadow Dragon and the Blade of Light. This mode was officially given a name in Fire Emblem: New Mystery of the Emblem in Japan and in Fire Emblem Awakening for international audiences.
In most tactical role-play games, if a character loses all of their HP, they are incapacitated for the rest of the battle or units can be replaced if there are permanent unit deaths. The Fire Emblem franchise operates differently from both of these features. Should a playable character lose all of their HP, that character dies and is permanently unavailable for usage in subsequent chapters after their death. In most Fire Emblem games, with the exception of Fire Emblem: Shadow Dragon, the games contain a limited roster, meaning that losing too many units can severely hinder a player's progress.
Losing most units does not directly stop a chapter. 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: New Mystery of the Emblem where the permanent death feature could not be disabled.
Certain characters in the series, such as Seth in Fire Emblem: The Sacred Stones or Frederick in Fire Emblem Awakening, will not die and will continue to appear in subsequent chapters wherever they are scripted to appear after losing all of their HP in battle. However, their wounds from losing all of their HP leaves them permanently retired for the rest of the game. To solidify this, most Credit Rolls at the end of the game still gives them a Retired and does not show their character epilogues.
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, Valkyrie Staff, or Bifröst (appearing only in Fire Emblem: Shadow Dragon and the Blade of Light, Genealogy of the Holy War, and Fates respectively) can be used to revive a character. However these staves are usually limited to one per playthrough, having usually 1 usage, and requires a high weapon proficiency with staves.
New Mystery of the Emblem introduced Casual Mode which disables the permanent death feature, instead reviving defeated allies at the end of the chapter. Fire Emblem Fates introduced Phoenix Mode which revives a defeated ally at the end of the player's turn.