I remember playing this game when it released, and at some place when you stole a certain object, an invisible dude appeared behind you and started attacking you. I think he was wearing a full set of enchanted daedric armor. I know this is vague, but does anyone know what I'm talking about?

Are you thinking of http://uesp.net/wiki/Morrowind:The_Lady%27s_Ring?

no, not that one. I'm pretty sure the quest (if it's even a quest) takes place in a house

You may have picked up one of the various cursed items in the game. They summon a Dremora Lord behind you when taken, but they aren't invisible or use chameleon.

Maybe the OP thought that the Dremora Lord had previously been invisible.

Yes... Possibly a Dremora (Lord) after stealing a certain item. There are many of these items, most of them in Daedric Shrines. But in a house? Well, there is particularly http://www.uesp.net/wiki/Morrowind:Daedric_Sanctuary_Amulet#Daedric_Sanctuary_Amulet. Might be some in some houses I'm not familiar with.

Another thing that comes to mind is a Dark Brotherhood assassin who actually wears unenchanted Dark Brotherhood armor, though, and like Dremora, aren't invisible before. They just appear out of nowhere.

One of the Ienith brothers appears behind you when you attack the other one in their room at the Dren plantation.
Don't think either is wearing armor though.

are you running jvk? Could have been the assassin : P

IIRC the Giants mod adds a humanoid, invisible creature called "night stalker" or "shade" or something like that, which sneaks up and attacks you at night. Not part of a quest, though, or in a house. Not sure.