A feature in many older (and some newer) first- and third-person shooters, auto-aim allowed for players not to have to fire exactly on target in order to hit what they're shooting at. With today's gaming technology, auto-aim usually gives players an unfair advantage over their opponants but this wasn't always the case: The classic Doom, for example, didn't allow the player to change the view vertically, as though the player character were looking up or down. This would not have been too much of a problem except for the fact that there were stairs, ledges, and other higher or lower altitude places enemies could situate themselves on in relation to the player's position. There were even some enemies that floated through the air. To make up for the inability of the player to aim up or down, auto-aim was used.
Even when some games (such as Heretic) added the ability to look up and down, auto-aim was retained due to difficulty in getting aim exact in some games. There were other things that gradually aided in removing the need for auto-aim (such as adding crosshairs) but the main device that seems to have allowed for greater accuracy by the player without the aid of auto-aim is the mouse. Of course, there were mice on computers since before the first- and third-person shooter genres came about but they didn't have much usefulness in some of the early games. I don't know about you but I've found trying to play an old game like Doom pretty damned annoying with the mouse. More modern (and with three-dimensional creatures instead of two-dimensional sprites) games, on the other hand, are much harder to play without a mouse.
Still some games have kept auto-aim as at least an optional setting (such as Half-Life) or as something always on because of how the game is designed: Max Payne has a slight auto-aim on (which decreases on the higher skill levels), presumably to make up for the slight awkwardness many third-person shooters have compared to first-person shooters, even with aids such as a crosshair. Return to Castle Wolfenstein is another example of a modern game with a slight auto-aim feature built-in. I assume this is because many of the weapons aren't as precise as they are in other games (this is supposed to be more realistic) and it's not easy for the player to compensate for this (though it would be nice if this was disabled for the hardest skill level). Multiplayer-centric first-person shooters, such as Starsiege Tribes or Quake III Arena, typically don't have auto-aim to increase the game's challenge and so more skilled players can actually make use of their increased accuracy.