What happens to the enemy's health after you die?

For normal enemies, including champions and rare mobs, damaged health will not return (assuming they have no self-healing abilities in the first place). On higher difficulties, however, they will regen, slowly (though dead champs stay dead, thankfully!)

For bosses, dying means they will reset to phase 1 and must be beaten starting from the beginning again. (Cinema scenes will replay too).

Dying while in a party is similar. Unlike in regular combat, you will not be able to revive until the boss is dead, and while losing a party member in the middle of a boss fight might make it harder, the boss won't reset until all of the party has been slain.

In later difficulties (definitely inferno, I don't recall hell), champs will regen life when no one is in combat with them. They might even do it while in combat, but if so I haven't noticed that. If you have to run across the map it's not uncommon to find mobs fully healed. Kite them back to your spawn place to minimize such healing.