to make atonement: to set right the damage that sin or impurity has done, by a ritual, a payment, or an equivalent life, so that the offender and the holy place are cleared and God's anger turned away.
This is one of the most loaded words in the Hebrew Bible, and the writers use it to do several jobs at once. At its core it names whatever it takes to undo the harm that wrongdoing or uncleanness has caused, so the guilty party is set free of the consequence and the relationship with God is restored. The act can be a sacrifice, a payment, or, in the gravest cases, a life given in place of another.
Four shades run through its uses. In the priestly ritual it is the technical word for the rite that cleanses the offerer and scours the sanctuary of the stain that sin leaves on it, the rite that climaxes once a year on the Day of Atonement. In ordinary speech it can mean to soothe or pacify someone's anger, the way a gift calms a person who has been wronged. It can name a ransom, an equivalent given so that a forfeited life is not taken, which is why for the worst crimes no payment is allowed and only blood will do. And in the prophets and wisdom books it is simply what God does when he wipes a sin away and forgives.
These are not four unrelated ideas but one act seen from different angles: something stands between the offender and God, and this word names whatever removes it. A sacrifice removes it by cleansing, a gift by calming, a ransom by paying, forgiveness by canceling. The same rite can do more than one of these at once, which is why no single English word captures it.
Scholars still argue over which shade is the original and the deepest. The older idea that it first meant to cover sin has been largely abandoned; most now think it first meant to wipe or scour clean, while others give first place to the idea of a ransom. The senses below lay out the range the word actually covers in the text, leaving that origin question to the disputes.