If one of your units loses all HP in battle, that character will perish and be removed from your party forever. Not only that, but any items the unit was carrying will be lost as well. If Hector, Eliwood, or Lyn loses all HP, your game will be over.

The only exception to this rule is Lyn's Story. Any characters who fall in battle during Lyn's Story will be lost until Chapter 11 (the first chapter of Eliwood's Story). However, you will have missed out on valuable experience points for that unit.