The Burnt Offering
The burnt offering (Leviticus 1) was so called because it was to be consumed by fire before God. It was not allowed to be eaten either by the priests or the offerer. According to the 1906 Jewish Encyclopedia, it was the highest order of Old Testament sacrifices.
"And the priest shall burn all upon the altar, a burnt sacrifice, an offering made by fire, of a sweet savor to the Lord." (Leviticus 1:9, HBFV throughout).
The offering symbolized the complete surrender or total dedication to God by an individual or a congregation. The sacrifice was always to be a male animal as indicating strength and energy. The animal was either a young bullock, a young ram or goat, a turtledove or young pigeons.
The blood of the sacrifice was thrown on the angles of the temple's altar. Then 'the sinew of the thigh' (Genesis 32:32), the stomach and the entrails, and so on, having been removed (in the case of birds also the feathers and the wings), and the sacrifice having been duly salted, it was wholly burned.
The skins of the burnt offering belonged to the ministering priests who derived a considerable revenue from them. It was the only sacrifice which non-Israelites were allowed to bring before God.
Burnt offerings were a part of the regular morning and evening service in Jerusalem's Temple. On Sabbaths, new moons, and festivals additional burnt offerings followed the ordinary worship.
Sin Offering
The sin offering (Leviticus 4) symbolized atoning for disobedience before God. Its meaning also represented the imperfection of our physical existence and the sinfulness of our human nature.
This sacrifice applied to sins repented of which were committed "through ignorance" or unintentionally (Numbers 15:22 - 23). It did not apply to sins performed willfully through full knowledge of what was being done (a presumptuous or "high handed" sin - Numbers 15:30). It also was applied to sins of weakness, or those committed where the offender, at the time, did not realize his guilt.
The sacrifice itself, according to the 1906 Jewish Encyclopedia, was the following.
"If the anointed priest or the whole congregation commits a sin through ignorance, the sin offering is a young bullock without blemish. Should the ruler so sin, his offering is a male kid without blemish.
"But when a private individual sins, his offering must be either a female kid or a female lamb without blemish, or, if he is too poor to provide one of these, a turtledove" (article on the sin offering).
On the Feast of Atonement, wherein all Israel fasted before God, the High Priest began the festival with two sin sacrifices. The first, a bullock, was for his own sins (and those of his household) while the second, a male kid of the goats, was for all the transgressions of the people (Leviticus 16:3, 5 - 16).
In ancient Israel a woman, forty days after giving birth to a male or eighty days for a female, was required to bring both a burnt (a lamb of the first year) and sin (a young pigeon or turtledove) offering to the temple (Leviticus 12). If the woman was poor and unable to afford a lamb, she could buy two pigeons or two turtledoves to cover these sacrifices. This is precisely what Mary and Joseph did forty days after the birth of Christ (Luke 2:22 - 24).
Sin offerings were also given in purification ceremonies after a male (likely) contacted gonorrhea (Leviticus 15:13 - 15) and after a person was healed of leprosy (14:2 - 19). Those who took the vow of a Nazarite were commanded to give sin and other offerings at the end of their service (Numbers 6:1 - 21).
Trespass Offering
Trespass offerings are generally discussed from Leviticus 5:1 to 7:7. They were given for acts committed through ignorance, or unintentionally, or when a person felt guilty that they may have sinned. It was also given by those who felt themselves in a special situation requiring this sacrifice. The act symbolized reconciling a person to God for their behavior.
The healing of a leper (Leviticus 14:12) or when a Nazarite had his vow interrupted by defilement through a dead body (Numbers 6:10 - 12), also called for this sacrifice.
The trespass offering itself was either the sacrifice of a ram or a male lamb. If the sin involved was "in the holy things of the Lord" (Leviticus 5:15), such as accidently not paying full tithes, the restitution was sometimes referred to as a guilt offering. In such cases the person was required to sacrifice a ram without blemish along with a certain amount of silver, payable through the sanctuary shekel, as restitution (Leviticus 5:14 - 19).
Peace Offering
According to the 1906 Jewish Encyclopedia, there are three types of peace offerings. They are thank offerings (Leviticus 7:11 - 12), votive (vow) offerings and free-will offerings (Leviticus 7:16). The thank offering is in acknowledgment of receiving an act of God's grace. The free-will offering is an act of love and generosity toward the Eternal.
Peace offerings, in general, symbolized complete peace with God. It was always accompanied by a meat and drink offering (Leviticus 7:11). Its primary focus was the sacrificial meal.
The two lambs "of the first year" given as a sacrifice at every Pentecost celebration (Leviticus 23:19) were a public peace offering that were eaten only by the officiating priests. King Solomon, when the temple was completed and the Ark of the Covenant was moved into the Holy of Holies, offered in sacrifice 22,000 oxen and 120,000 sheep as a peace offering to God (1Kings 8:63).
Conclusion
The Old Testament system of offerings at Jerusalem's temple was at the center of worshipping God. Such gifts, however, were abolished in the New Testament under the New Covenant (Hebrews 9:9 - 10, 10:1 - 18). The fullness of their meaning was fulfilled by the perfect sacrifice of Jesus Christ for our sins (Hebrews 9:12).