My friend and I were just playing Diablo 3 on Xbox One when we heard a weird sound in the Act I Spider Cave.
Then out of nowhere a red and black demon came out and ran for my friend! The name over the demon was “Hunter of (My friend’s character name)”. We had no idea what was happening and after my friend was killed the demon left? We have never seen that happen and I don’t know if that was a Demon Hunter (my friend’s character) thing or not?
This is the Nemesis System. It is unique to the console version of the game.
From the Blizzard Diablo III Console page FAQ:
Nemesis System: If a monster defeats you, it has a chance of invading the games of players in your friends list! Nemesis monsters that originate in a friend’s game may also invade your own game.
From the Diablowiki page for Diablo III:UEE:
The Nemesis System can take a monster that kills one player and make that monster appear, with added powers, in the game of someone on the friend’s list. If the monster kills that player, it will go to another person on his Friend’s List and so on, up to five times.