Read Leviticus 5:14 -6:7; 19:20-22; Numbers 5:5-8; Ex 22:1-15; Lev 14, 22:14-16.
The trespass offering was most often used when there could be a monetary value assessed to the damage done, and where the law broken was not so fundamentally important for the well being of the victim.
In the sin offering the substitutionary animal bore the full penalty of the offer's sin, whereas in the trespass offering restitution was needed to satisfy G-d's justice. Restitution repaired the damage to the victim, but this alone could not satisfy G-d's broken law, so an offering was needed as well.
Only a ram could be offered for the trespass offering. There was not a difference in the offering whether the offerer was poor or rich. This taught that the consequences of ones actions are not altered by ones circumstances. G-d is not an advocate of situation ethics.
Sacrifices played an important part in Israel's judicial system. Depending on the crime, a sin or trespass offering was offered, although both were offered for sins, which broke no civil law. This was calculated to teach that crime was not only against a neighbor, but G-d too, by dishonoring His laws given on behalf of the nation. There was always a 1/5th added to the restitution. If the offerer was only to repair what he had done he would have lost nothing. By paying an extra 1/5 he learned that sin never profits.
Trespasses against G-d are identified as those pertaining to holy things (Lev 5:15) (ie. Failure to pay tithes)
Trespasses against man could include a variety of offenses. (Lev 6:2-3) (ie. Lying)
The trespass offering dealt mainly with physical injury or cases where a monetary value could be assessed for damages.
When the offering was made the rump, and all the fat that covers the entrails and the lobe of the liver, and the two kidneys and the fat that is on them where burned on the brazen alter.
The priests ate the remaining parts. The offerer did not have any portion. The blood was sprinkled on the altar. Restitution was always made before the offering.