There were different types of offerings and sacrifices made for other reasons, but animal sacrifices made for atonement are the ones that kept the priests busy. This is the very beginning of “if this happens, do this” that is prescribed by God in the Law.

Leviticus 4

The LORD spoke to Moses, saying:

2 “Speak to the children of Israel, and say ‘If a person sins through ignorance against any of the commandments of the LORD concerning things which ought not to be done, and does any of them:

3 If the priest that is anointed sins and brings guilt on the people; then let him bring for his sin a young bull without blemish to the LORD for a sin offering.

4 He shall bring the bull to the door of the tent of meeting before the LORD; and shall lay his hand on the bull’s head, and kill the bull before the LORD.

5 And the priest that is anointed shall take some of the bull’s blood, and bring it to the tent of meeting:

6 The priest shall dip his finger in the blood, and sprinkle some of the blood seven times before the LORD, in front of the veil of the sanctuary.

7 And the priest shall put some of the blood on the horns of the altar of sweet incense before the LORD, which is in the tent of meeting; and shall pour all the blood of the bull at the bottom of the altar of the burnt offering, which is at the door of the tent of meeting.

8 And he shall take from it from the bull all the fat for the sin offering; the fat that covers the entrails, and all the fat that is on the entrails,

9 the two kidneys, and the fat that is on them, which is by the loins, and the long lobe of the liver, with the kidneys, shall be removed. 

10 just as it is taken from the bull of the sacrifice of peace offerings: and the priest shall burn them on the altar of the burnt offering.

11 And the skin of the bull, and all his flesh, with his head, legs, entrails, and dung,

12 the whole bull he shall carry out of the camp to a clean place, where the ashes are poured out, and burn him on the wood with fire: where the ashes are poured out he shall be burnt.

13 If the whole congregation of Israel sin through ignorance, and the thing is hid from the eyes of the assembly, and they have done something against any of the commandments of the LORD concerning things which should not be done, and are guilty;

14 When the sin becomes known, then the congregation shall offer a young bull as a sin offering and bring him before the tent of meeting.

15 The elders of the congregation shall lay their hands on the head of the bull before the LORD: and the bull shall be killed before the LORD.

16 And the priest that is anointed shall bring some of the blood to the tent of meeting:

17 The priest shall dip his finger in some of the blood, and sprinkle it seven times before the LORD, in front of the veil.

18 He shall put some of the blood on the horns of the altar which is before the LORD, that is in the tent of meeting, and shall pour out all the blood at the bottom of the altar of the burnt offering, which is at the door of the tent of meeting.

19 He shall take all his fat from him, and burn it on the altar.

20 And he shall do with the bull as he did with the bull for a sin offering, he shall do the same with this: and the priest shall make atonement for them, and it shall be forgiven them.

21 He shall carry the bull out of the camp, and burn him as he burned the first bull: it is a sin offering for the congregation.

22 When a ruler has sinned, and done something through ignorance against any of the commandments of the LORD his God concerning things which should not be done, and is guilty;

23 Or if his sin comes to his knowledge; he shall bring his offering, a kid of the goats, a male without blemish:

24 And he shall lay his hand on the head of the goat, and kill it in the place where they kill the burnt offering before the LORD: it is a sin offering.

25 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 shall pour out his blood at the bottom of the altar of burnt offering.

26 He shall burn all his fat on the altar, as the fat of the sacrifice of peace offerings: and the priest shall make atonement for him concerning his sin, and it shall be forgiven.

27 If any one of the common people sin through ignorance, and does something against any of the commandments of the LORD concerning things which ought not to be done, and be guilty;

28 Or if his sin comes to his knowledge: then he shall bring his offering, a kid of the goats, a female without blemish, for the sin which he has sinned.

29 He shall lay his hand on the head of the sin offering, and slaughter the sin offering in the place of the burnt offering.

30 The priest shall take some of the blood with his finger, and put it on the horns of the altar of burnt offering, and shall pour out all the blood of it at the bottom of the altar.

31 And he shall take away all the far, as the fat is taken away from the sacrifice of peace offerings; and the priest shall burn it on the altar for a pleasing aroma to the LORD; and the priest shall make atonement for him, and it shall be forgiven.

32 If he brings a lamb for a sin offering, he shall bring it a female without blemish.

33 He shall lay his hand on the head of the sin offering, and slaughter it for a sin offering in the place where they kill the burnt offering.

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 shall pour out all the blood of it at the bottom of the altar:

35 And he shall take away all the fat, as the fat of the lamb is taken away from the sacrifice of the peace offerings; and the priest shall burn them on the altar, according to the offerings made by fire to the LORD: and the priest shall make atonement for his sin that he has committed, and it shall be forgiven.'”

See also: Leviticus 4 in ESV, HCSB, NASB and NIV via Bible Gateway.