"Be on your guard! If your brother sins, rebuke him; and if he repents, forgive him. And if he sins against you seven times a day, and returns to you seven times, saying, 'I repent,' forgive him" (Luke 17:3-4).
Sin occurs when someone breaks the law. Many of the laws are about how we treat our neighbors. "For this, "You shall not commit adultery, you shall not murder, you shall not steal, you shall not covet," and if there is any other commandment, it is summed up in this saying, "You shall love your neighbor as yourself." Love does no wrong to a neighbor; therefore love is the fulfillment of the law" (Romans 13:9-10). Thus, when someone steals, he sinning by breaking the law against stealing, but he is also acting against his neighbor.
Therefore, people can commit sins that are against you.
To forgive is to say that someone is not indebted to you -- they don't owe you anything. When a person sins against a neighbor, they have caused harm in some way. They might have stolen money from a neighbor, harmed his reputation, or wounded his feelings. When the sinner changes and turns from his sins, those harmed by his sin are required to no longer hold the former sinner under obligation to them.