What does "Jesus died for our sins" mean, theologically?

This being GQ, you can tell I want a factual answer: what exactly do people mean when they say “Jesus died for our sins”? Whose sins? Everyone who lived after him? Everyone everywhere, at any time? Were sins just lessened, or eliminated entirely? Was his death a “clean slate,” or something different?

Best thing to do by way of factual answer, then is to send you to Wikipedia on Atonement which seems to be the most comprehensive summary of what it means in different interpretations.

You have to understand that there is no single universal school of though on this; different religious bodies and even factions and individuals within them can all be saying essentially that and be meaning different things.

But in my experience they mean this:
Jesus took upon himself the punishment and/or consequence of our sin, including both our inherent state of sinfulness (or disfellowship with God) - inherited as a consequence of being a mortal, fallen human descended from a mortal, fallen progenitor(this is also called ‘original sin’), and our individual acts of personal sin that we commit through wickedness and neglect.
It means that the penalty of those sins was visited in full upon Jesus and these penalties/consequences can therefore be removed from us by our acceptance of him as lord (in the case of original sin) and true repentance (in the case of personal sin).

I’m not going to try to defend any of these viewpoints, I’m just trying to describe them as they exist in the mindset of Protestant Christianity.

The whole “Jesus dying for our sins” goes back a long way. It is just an evolved version of something called substitutionary atonement, which can be seen in the Judeo-Christian tradition back in the Old Testament. Therein, the faithful would transfer all of their sins to a goad and set the goat to wander in the desert. This, literally, was a scapegoat. I’m sure that other traditions use this concept, but my background in theology is limited to the courses I had to take at my prep school. The Jesus figure, in a twist on the old scapegoat, actually had the sins of humanity transferred to him and he was killed in order to atone for these sins. That is to say that one is able to attain salvation because one’s sins have already been atoned-for. One needs merely to believe that this is the case. Or so goes the Christian tradition. The validity of that, of course, belongs over in GD but this is the doctrine as put forth by most mainstream religions.

In a nutshell, it’s the major product packaged and sold by Christianity. What they tell you is that if you join up with them and give them money, they will let you believe in this guy named Jesus who supposedly took one on the chin for you 2000 years ago. But if you don’t pony up the bucks, you don’t get to believe in him and you’ll suffer for it.

It’s the basic model of any religion with the blanks filled in.

That interpretation actually only dates back to Anselm, in the Middle Ages. The far older tradition, and the one still held to in the Eastern Church, is that Christ died and then rose again to remove the effects of sin, i.e. death and corruption, rather than guilt, which can only be removed by repentance.