Text (LG+errata): At the beginning of each player's upkeep, destroy target nonartifact creature that player controls of his or her choice. It can't be regenerated. [Oracle 2001/08/24]
This is a targeted ability. The Abyss targets one creature each upkeep. It cannot target creatures which are untargetable, such as those with Protection from Black. [Aahz 1994/07/13]
If the target becomes invalid after being chosen but before resolution, you do not have to choose another target. [Duelist Magazine #16, Page 25]
The controller of The Abyss controls the ability, but the current player chooses what gets destroyed. [Barclay 2000/12/31]
Extended tournaments (see Rule 913) have always banned this card.