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.