"When his soul makes an offering for guilt," we read in Isaiah 53:10, part of Isaiah's famous Suffering Servant prophecy. The word for "offering for guilt" is asam, the name for one of the kinds of sacrifices offered at the tabernacle or temple.
Scholars today, following Jacob Milgrom, often refer to this type of sacrifice as a reparation offering. Most English translations use "trespass offering" (KJV) or "guilt offering" for this kind of offering, but there are some variations. CEB uses "compensation offering" and HCSB uses "restitution offering."
In chapter 3 of The Gospel According to Isaiah 53, Walter Kaiser notes that a reparation offering is usually offered by an individual and makes sure that a debt is paid in full. These qualities are appropriate for the sacrifice offered by the Messiah for the sins of mankind.
Reparation offerings were called for in cases of "desecration of sancta"---that is, when something holy had been mishandled or mistreated (Lev 5:14-6:7). One example would be a situation where someone accidentally ate food that had been set aside for the Levitical priests (Lev 22:14). In that case, one would bring a reparation offering and make reimbursement for the food, adding a fine of 20 per cent.
Another example was a situation where a Nazirite accidentally came in contact with a dead body (Num 6:9-12). In that situation the holy thing that had been defiled was the Nazirite's hair, which was being prepared for an offering at the completion of the vow.
A third example was a situation where one person robbed another and then came to regret it and wanted to make things right (Lev 6:1-7). Here the thing that had been violated was a fellow Israelite, one of God's holy people. In this case the stolen amount was restored along with a 20 per cent fine, and a reparation offering was carried out.
A fourth example was the reinstatement to the Israelite community of a person who had been suffering from a skin disease. The skin disease had desecrated a member of God's holy nation. The ritual carried out in this case (Lev 14:12-18), which involved sprinkling of blood, restored the person to the "kingdom of priests."
More broadly, Israel's exile was a desecration of Israel. The Servant offered himself as a reparation offering so that Israel could be restored to the holy status of God's servant.
In Isaiah 53:8, God says that the Servant was "stricken for the transgression of my people." The word for "stricken" appears 78 times in the Old Testament, 61 of them in Leviticus 13-14, the chapters on skin diseases. The prophecy pictures the Servant becoming a "leper" for the sake of the people. Its references to the marred appearance of the Servant (52:14; 53:2-3) are consistent with such a picture.
Another detail that could tie in is in Isaiah 52:15, which says that the Servant will "sprinkle many nations." As mentioned above, the ritual for restoring a person healed of a skin disease involved sprinkling of blood. Overall, the prophecy pictures the Servant as both leper and offering for the leper, making atonement for "leprous" nations.