Ok here is my stab at this rather important question which goes deep into the Christian faith.
Let me start with these verses:
Romans 6:23 New King James Version (NKJV) For the wages of sin is
death, but the gift of God is eternal life in Christ Jesus our Lord.
Romans 5:12 New King James Version (NKJV) Therefore, just as through
one man sin entered the world, and death through sin, and thus death
spread to all men, because all sinned
and
"without shedding of blood is no remission of sins" (Heb 9:22, 26).
These verse set the premise that it is sin that requires death in order for it to be 'satisfied'.
(Some more verses to study: Eze. 18:4; Heb 10:3-4; Rom 5:11)
In the Old Covenant, i.e., The Old Testament, the Israelites were given the Law by Moses to fulfil two things:
- Prescribed the technique that should be employed in order to permit a way to temporarily deal with their sin (I will talk more on this further down).
- The other purpose, and many will argue the more important reason, is to show the Israelites where they were going wrong; it was not meant to be a permanent system to deal with their sins. It was there to make it obvious they could never meet the high standard of God. Romans deals with this whole question really but look at Romans 3 and especially verse 23 for this point.
Ok, talking about dealing with their sin, my first point. The sacrifices they made were made so that the blood of their sacrifices would make the remittance of sin possible:
"without shedding of blood is no remission of sins" (Heb 9:22, 26).
This is actually a symbol of Jesus, which is partly why we call him the lamb, the worthy sacrifice. The point that needs to be made is by the first scripture (Romans 5:12) '...one man sin entered the world, and death through sin, and thus death spread to all men, because all sinned...", the first man being perfectly created namely Adam and the second perfect man Jesus. The Sin Offering was a temporary fix but it wasn't until the perfect man was sacrificed by His father that the permanent fix could be established.
You see, doing works of the law could not possibly justify:
Romans 3:20 New King James Version (NKJV)
Therefore by the deeds of the law no flesh will be justified
in His sight, for by the law is the knowledge of sin.
Hebrews 9:9 New King James Version (NKJV)
It was symbolic for the present time in which both gifts and
sacrifices are offered which cannot make him who performed
the service perfect in regard to the conscience-
It is only by grace through faith that we are able to receive the justification and atonement to regain our place as people of God.
Ephesians 2:8-9 New King James Version (NKJV) For by grace you have been
saved through faith, and that not of yourselves; it is the gift
of God, 9 not of works, lest anyone should boast.
The word 'works' actually means works of the law it should not be confused with James 2:17 "...Thus also faith by itself, if it does not have works, is dead.", because that is talking about an inward change being manifested outwardly in our changed attitudes to serve our brothers, sisters and communities to the glory of God.
So, the answer to your question is that when sacrifices were made they had to be with a faithful heart, with the desire to honour God. Take Abel and Cain, Abel was said to have offered a sacrifice that pleased Him and in Hebrews 11 it says his offering still speaks today. When a sacrifice was made it was with utmost reverence!
The word 'atonement' has been told a 100 times to many a bored sunday school kid to be the same as at-one-ment, to be at one with God. The whole purpose of the human race is to be with God like Adam and Eve in the garden, walking with him and having relationship with Him. that is what atonement is trying to achieve, and it is fully and completely achieved by Jesus dying on the cross being the propitiation, that is suffering the wrath of God on every ones behalf because, if you remember, the wages of sin is death.
I hope this helps you get what you are after, or at the very least aid someone else in trying to answer this question.
Here is a list of the verses that talk specifically about Sin Offerings:
http://www.biblegateway.com/passage/?search=Leviticus+4&version=NKJV
http://www.biblegateway.com/passage/?search=Leviticus+5&version=NKJV
http://www.biblegateway.com/passage/?search=Leviticus+6&version=NKJV
http://www.biblegateway.com/passage/?search=Leviticus+8&version=NKJV
http://www.biblegateway.com/passage/?search=Leviticus+16&version=NKJV
Please read the whole of the chapters to ensure it is kept in context.