I can only give an overview in this answer. The details and proof are such that it runs to ten pages which simply cannot be reproduced here.
LINK : The full explanation, containing all references and texts, is available from my website as a PDF to view online HERE or to download.
The word has been used to translate two Hebrew words, kaphar and kippurim. Little notice has been taken of the fact that kippurim is a plural word. Attempts have been made (particularly by means of Masoretic 'pointing') to add an extra 'p' to kaphar to make it look as though it is the same word as kippurim, which it is not.
'To make an atonement' is not a satisfactory way to translate a verb. Kaphar is a verb and has a meaning all of its own. To translate with an infinitive and a noun phrase is just unscientific.
The same kind of treatment is true of the word 'redemption' which is used to translate both gaal and padah in Hebrew and used to translate both the lutron group (lutrosis/lutrotes/apolutrosis) and the agora group (agoratso/exagoratso) in Greek. This is just unscientific, they are different concepts.
Atonement supposedly means, quite literally, 'at-one-ment', a vague concept of (perhaps) "unity by reconciliation". It has been used in the KJV, once, to translate katallage which usually is translated 'reconciliation'. But this is clearly inappropriate.
Kaph means hand. Or the sole of the foot. It is not the working hand or fighting fist, which is expressed by yod. Kaph is the cupped hand such as is used to convey water to the mouth. It is a containment.
Following the word kaph through Genesis reveals what kaph conveys spiritually. Pharaoah's integrity and Jacob's afflictions are all gathered up in the nuance of the word as its use develops through scripture. Cleanness of hand and sensitivity of purpose combine in the meaning of kaph.
On Horeb, God's kaph protected Moses in the cleft of the rock as Deity passed by, that he might not perish. Moses was contained within the personal hand of God from what Deity is by nature in relation to fallen humanity.
The noun form, kopher, describes a fenced village. A misunderstanding has caused a mistranslation and the progression of settlement/fenced village/city/citadel has been disrupted because it was not appreciated that the chaotic lifestyle of Philistines led to their careless lack of fencing to protect their womenfolk and little ones.
Understood properly, kopher means a protected settlement. It is contained.
When Noah and his sons built the ark he was told to 'kaphar the gopher with kopher'. Gopher is not a botanical description - it means molten. Lava and pitch (gophrith) may both be described as 'molten'. Both result from judgement; both are high in sulphur and thus get inside the lungs and burn; both are very hot (when molten) and both burn the exterior skin.
The wood was to be steeped in molten (almost certainly pitch) first to penetrate the very fibres of the wood. Once steeped it becomes 'gopher' or 'gopher wood'. It does not refer to the botanical species it refers to the treatment of the wood. Only then was it smeared, inside and outside, with 'kopher'.
Kopher refers also to what was almost certainly pitch, but in a different way. The steeping is inward, the outer cladding is a matter of containment.
This is all a practical matter of waterproofing but it also conveys spiritual concepts which are only fully revealed in the New Testament scriptures consummate upon the sufferings, death and resurrection of Jesus Christ. Part of what occurred at Golgotha is a matter of containment. He was made sin. Sin was contained.
This containment is seen right at the beginning of scripture when Adam and his wife were clothed with skins. The skin of another was about them. They were contained within the skin of another.
I am crucified with Christ, says Paul. Nevertheless he lived.
Purah is foliage. It is the 'rush' (branch and rush); twigs and leaves, basically. Kippah is the main branch as used in scripture. Kippurah - the entire branch with foliage - is a word never used in the Hebrew scriptures. Kippurim is its plural form. It means (living) 'branchings'. It has immense spiritual implications. The kippurim concept and the kippurim word are not the same as that of kaphar. They are different words and they are different concepts.
Kaphar is a matter of containment and kippurim is a matter of 'branchings'. They are different concepts and both are expressed in the Greek scriptures as different concepts. I am the Vine, ye are the branches conveys one concept. He was made sin conveys another.
Branchings and containment.