This is a great question. The two main western religions are Judaism and Christianity. When we look at these two religions we can say a lot about sin and forgiveness.
First, Judaism makes a very important theoretical distinction between moral sins and impurity. Moral sins can be defined as anything morally inconsistent with the character of God as revealed in the Bible. A summary of this can be seen the Ten Commandments. As for impurity, it is really in the realm of rituals. For example, if a priest touches a dead body, then that person is deemed unclean. Another example would be the impurity of a women after giving birth to a child. For a time, she is deemed unclean and through rituals, she becomes clean again. There is nothing moral about this. Forgiveness is found in both of these cases mainly in the sacrificial system. Most of these rules are found in the book of Leviticus.
When it comes to Christianity, there is a shift. Sin is almost exclusively thought to be seen in moral terms. Forgiveness is found not through animal sacrifices, as in the Hebrew Bible, but in the sacrifice of the second person of the Trinity. Moreover, it is faith in Jesus that makes forgiveness take place.