SIN and SIN-OFFERING in the Old Testament are both translated from the same Hebrew word, chatta'ath. Every time in the Old Testament where the word is used, the translators needed to use context to determine whether to translate it as SIN or as SIN-OFFERING. This is odd, because we tend to define SIN as "bad thing" and SIN-OFFERING as "good thing." SIN is the offense, and the SIN-OFFERING is price paid for that offense – they appear to be opposites.
What deepens the complication is when translators disagree on the context.
Take for example the first time the Hebrew word, chatta'ath, is used. In Genesis 4, after Cain fails to offer a correct sacrifice, God speaks:
Then the Lord said to Cain, “Why are you angry? Why is your face downcast? If you do what is right, will you not be accepted? But if you do not do what is right, sin is crouching at your door; it desires to have you, but you must rule over it.”
~ Genesis 4:6-7 (NIV)
The New International Version (NIV) Bible translates chatta-ath as SIN in this passage, as do many of the modern translations. When read this way, and pushed forward as doctrine, we have an image of a personal evil presence at our door, which crouches, desiring to have us, but we are instructed to rule over it.
This is a very common way of looking at the relationship between people and evil: evil is always in front of us, but we have to make good choices to stay in God's good graces. We must rule over any evil desires that present themselves.
In contrast, Young's Literal Translation renders it this way:
And Jehovah saith unto Cain, `Why hast thou displeasure? and why hath thy countenance fallen? Is there not, if thou dost well, acceptance? and if thou dost not well, at the opening a sin-offering is crouching, and unto thee its desire, and thou rulest over it.'
~ Youngs Literal Translation 4:6-7 (YLT)
If you can get past the old-English use of “thee” and “thou,” you'll notice that the YLT translators use SIN-OFFERING instead of SIN. If you understand SIN-OFFERING to be a lamb (in a nod to the sacrifice of the Lamb of God, Jesus), the image is not of an evil force that is crouching, waiting to pounce, but rather of a personal willing sacrifice, in the form of a lamb, physically waiting at Cain's door – an offering that God would accept alongside Abel's offering.
This carries such a different meaning, and provides much more grace than the SIN translation. When translated as SIN, the responsibility for maintaining God's approval lies on our shoulders. We must make good choices and rule over any temptation and sinful desire. But when translated as SIN-OFFERING, we have a picture of God Himself making a way out; the work of grace is made available at our door. God Himself provides the lamb.
So how do we reconcile the complete differences in meaning? Who is “right” in their translation?
I believe that a proper understanding comes from another way of looking at this Hebrew word. Chatta'ath perhaps should be read as “consequence of the offense.” Rendered that way, each instance of the word, whether translated as SIN or as SIN-OFFERING is quite clear: there is a consequence for disobedience. Either we carry that consequence ourselves, or the sacrifice carries it away from us.
When the New Testament says that Jesus “became sin” for us (2 Corinthians 5:21), there's some depth to that statement that is often missed in the English. (It is important to note that the Greek text uses a word for SIN [hamartia] that can only be translated as SIN, or OFFENSE. Some translations render this as trespass. But there is no dual meaning like there is with the Hebrew chatta'ath.)
Perhaps Paul, writing this letter to the Corinthians, understood that the Hebrew scriptures described a sacrifice that was necessary to cover the consequence of Adam's disobedience, and that Jesus Himself was that sacrifice. The sacrifice on the cross described both what Jesus did as well as who he was: the covering over Adam in Genesis 3, and the appropriate offering to God in Genesis 4, both provided by God Himself and acceptable to Him.