The Elder Scrolls V: Skyrim – Do Companions Use Poisons?

Follower death is handled pretty oddly;

• If their health is exhausted due to enemy fire, they just get knocked out. Healing them or finishing combat and allowing them to regenerate will bring them back to life.
• If you kill them due to friendly fire, that's it: they're dead permanently.

But there are situations that may cause a follower to die permanently, even after they enter the "knocked out"/"cower" state: if an enemy keeps wailing on them, there's a chance they'll die permanently.

But it's not guaranteed: looking around online, I've found numerous accounts of Lydia or Faendal dying easily, but others who swear nothing they do kills them. UESP describes the follower death mechanics as such:

When your follower would die to an enemy attack or a trap, they will be incapacitated for some time and begin to recover their health, just like essential NPCs; enemies will stop attacking or targeting them during this time. However, if they lose all their health again during this period, they will die for good. Thus, you should try to move the fight away from them, so that they are not hit by a stray arrow or area-effect spell. They can also die if hit with a damage-over-time poison that is stronger than their health recovery. They can also die from enemy power-attacks.

requirements for completion:

• Only a melee weapon used (no ranged or spells)
• Weapon not enchanted