The Encyclopedia Britannica says:

Sacrifice, a religious rite in which an object is offered to a divinity in order to establish, maintain, or restore a right relationship of a human being to the sacred order. It is a complex phenomenon that has been found in the earliest known forms of worship and in all parts of the world.

A selection of verses showing the principles of sacrifice before and in the law of Moses.

After the flood, Noah offers a sacrifice that is a pleasant aroma to the LORD:

Genesis 8 verses 20 - 21

20 Noah built an altar to the LORD, and took of every clean animal, and of every clean bird, and offered burnt offerings on the altar. 21 The LORD smelled the pleasant aroma. The LORD said in his heart, “I will not again curse the ground any more for man’s sake because the imagination of man’s heart is evil from his youth. I will never again strike every living thing, as I have done.

After Moses receives the ten commandments, God tells him that they must make an altar to offer sacrifices:

Exodus 20 verse 24

24 You shall make an altar of earth for me, and shall sacrifice on it your burnt offerings and your peace offerings, your sheep and your cattle. In every place where I record my name I will come to you and I will bless you.

Moses is told the significance of the offerings:

Sin offering:

Leviticus 4 verse 2

2 “Speak to the children of Israel, saying, ‘If anyone sins unintentionally, in any of the things which the LORD has commanded not to be done, and does any one of them,

Leviticus 4 verses 34 - 35

34 The priest shall take some of the blood of the sin offering with his finger, and put it on the horns of the altar of burnt offering; and all the rest of its blood he shall pour out at the base of the altar. 35 He shall remove all its fat, like the fat of the lamb is removed from the sacrifice of peace offerings. The priest shall burn them on the altar, on the offerings of the LORD made by fire. The priest shall make atonement for him concerning his sin that he has sinned, and he will be forgiven. 

Trespass offering:

Leviticus 5 verses 15 - 16

15 “If anyone commits a trespass, and sins unwittingly regarding the LORD’s holy things, then he shall bring his trespass offering to the LORD: a ram without defect from the flock, according to your estimation in silver by shekels, according to the shekel of the sanctuary, for a trespass offering. 16 He shall make restitution for that which he has done wrong regarding the holy thing, and shall add a fifth part to it, and give it to the priest; and the priest shall make atonement for him with the ram of the trespass offering, and he will be forgiven. 

Leviticus 6 verse 2

2 “If anyone sins, and commits a trespass against the LORD, and deals falsely with his neighbour in a matter of deposit, or of bargain, or of robbery, or has oppressed his neighbour,

Leviticus 6 verse 6

6 He shall bring his trespass offering to the LORD: a ram without defect from the flock, according to your estimation, for a trespass offering, to the priest.

Burnt offering:

Leviticus 1 verses 3 - 4

3 “ ‘If his offering is a burnt offering from the herd, he shall offer a male without defect. He shall offer it at the door of the Tent of Meeting, that he may be accepted before the LORD. 4 He shall lay his hand on the head of the burnt offering, and it shall be accepted for him to make atonement for him.

Peace offering:

Leviticus 7 verses 15 - 16

15 The flesh of the sacrifice of his peace offerings for thanksgiving shall be eaten on the day of his offering. He shall not leave any of it until the morning. 
16 “ ‘But if the sacrifice of his offering is a vow, or a free will offering, it shall be eaten on the day that he offers his sacrifice. On the next day what remains of it shall be eaten,

Meal offering:

Leviticus 2 verses 1 - 2

1 “ ‘When anyone offers an offering of a meal offering to the LORD, his offering shall be of fine flour. He shall pour oil on it, and put frankincense on it. 2 He shall bring it to Aaron’s sons, the priests. He shall take his handful of its fine flour, and of its oil, with all its frankincense, and the priest shall burn its memorial on the altar, an offering made by fire, of a pleasant aroma to the LORD.