In ancient times, a major component of Jewish ritual was the offering of Karbanot. An entire order of the Talmud is devoted to the subject.
The word "Karbanot" is usually translated as "sacrifices" or "offerings"; however, both of these terms suggest a loss of something or a giving up of something, and although that is certainly a part of the ritual, that is not at all the literal meaning of the Hebrew word. The word Karbanot comes from the root Qof-Resh-Bet, which means "to draw near," and indicates the primary purpose of offerings: to draw us near to G-d.
Parts of the rituals involved in the offering of Karbanot were performed exclusively by the kohanim (priests). These rituals were only performed in the Temple in Jerusalem. The procedures could not be performed by anyone else, and could not be performed in any other place. Because the Temple no longer exists, we can no longer offer Karbanot.
There are three basic concepts underlying Karbanot. The first the aspect of giving. A korban requires the renunciation of something that belongs to the person making the offering. Thus, sacrifices are made from domestic animals, not wild animals (because wild animals do not belong to anyone). Likewise, offerings of food are ordinarily in the form of flour or meal, which requires substantial work to prepare.
Another important concept is the element of substitution. The idea is that the thing being offered is a substitute for the person making the offering, and the things that are done to the offering are things that should have been done to the person offering. The offering is in some sense "punished" in place of the offerer. It is interesting to note that whenever the subject of Karbanot is addressed in the Torah, the name of G-d used is the four-letter name indicating G-d's mercy.
The third important concept is the idea coming closer. The essence of sacrifice is to bring a person closer to G-d.
For the most part, the practice of sacrifice stopped in the year 70 C.E., when the Roman army destroyed the Second Temple in Jerusalem, the place where sacrifices were offered. The practice was briefly resumed during the Jewish War of 132-135 C.E., but was ended permanently after that war was lost. There were also a few communities that continued sacrifices for a while after that time.
Sacrifices were stopped after the Temple's destruction because the Torah specifically commands Jews not to offer sacrifices just anywhere; they are only permitted in the place that G-d has chosen for that purpose. It would be a sin to offer sacrifices in any other location.
Contrary to popular belief, the purpose of Karbanot is not simply to obtain forgiveness from sin. Although many Karbanot have the effect of expiating sins, there are many other purposes for bringing Karbanot, and the expiatory effect is often incidental, and is subject to significant limitations.
Certain Karbanot are brought purely for the purpose of communing with G-d and becoming closer to Him. Others are brought for the purpose of expressing thanks to G-d, love or gratitude. Others are used to cleanse a person of ritual impurity (which does not necessarily have anything to do with sin). And yes, many Karbanot are brought for purposes of atonement.
The atoning aspect of Karbanot is carefully circumscribed. For the most part, Karbanot only expiate unintentional sins, that is, sins committed because a person forgot that this thing was a sin. No atonement is needed for violations committed under duress or through lack of knowledge, and for the most part, Karbanot cannot atone for a malicious, deliberate sin. In addition, Karbanot have no expiating effect unless the person making the offering sincerely repents his or her actions before making the offering, and makes restitution to any person who was harmed by the violation.
There are many different types of Karbanot, and the laws related to them are detailed and complicated. This section introduces some of the major types of Karbanot - there are many subtypes within these classifications and other types that do not fit into these categories.
Olah: Burnt Offering
Perhaps the best-known class of offerings is the burnt offering. It was the oldest and commonest sacrifice, and represented submission to G-d's will. The Hebrew word for burnt offering is olah, from the root Ayin-Lamed-Heh, meaning ascension. It is the same root as the word aliyah, which is used to describe moving to Israel or ascending to the podium to say a blessing over the Torah. An olah is completely burnt on the outer altar; no part of it is eaten by anyone. Because the offering represents complete submission to G-d's will, the entire offering is given to G-d (i.e., it cannot be used after it is burnt). It expresses a desire to commune with G-d, and expiates sins incidentally in the process (because how can you commune with G-d if you are tainted with sins?). An olah could be made from cattle, sheep, goats, or even birds, depending on the offerer's means.
Zevach Sh'lamim: Peace Offering
A peace offering is an offering expressing thanks or gratitude to G-d for His bounties and mercies. The Hebrew term for this type of offering is zebach sh'lamim (or sometimes just sh'lamim), which is related to the word shalom, meaning "peace" or "whole." A representative portion of the offering is burnt on the altar, a portion is given to the kohanim, and the rest is eaten by the offerer and his family; thus, everyone gets a part of this offering. This category of offerings includes thanksgiving-offerings (in Hebrew, Todah, which was obligatory for survivors of life-threatening crises), free will-offerings, and offerings made after fulfillment of a vow.
Chatat: Sin Offering
A sin offering is an offering to atone for and purge a sin. It is an expression of sorrow for the error and a desire to be reconciled with G-d. The Hebrew term for this type of offering is chatat, from the word chayt, meaning "missing the mark." A chatat could only be offered for unintentional sins committed through carelessness, not for intentional, malicious sins. The size of the offering varied according to the nature of the sin and the financial means of the sinner. Some chatatot are individual and some are communal. Communal offerings represent the interdependence of the community, and the fact that we are all responsible for each others' sins. A few special chatatot could not be eaten, but for the most part, for the average person's personal sin, the chatat was eaten by the kohanim.
Asham: Guilt Offering
A guilt offering is an offering to atone for sins of stealing things from the altar, for when you are not sure whether you have committed a sin or what sin you have committed, or for breach of trust. The Hebrew word for a guilt offering is asham. When there was doubt as to whether a person committed a sin, the person would make an asham, rather than a chatat, because bringing a chatat would constitute admission of the sin, and the person would have to be punished for it. If a person brought an asham and later discovered that he had in fact committed the sin, he would have to bring a chatat at that time. An asham was eaten by the kohanim.
Food and Drink Offerings
A meal offering (minchah) represented the devotion of the fruits of man's work to G-d, because it was not a natural product, but something created through man's effort. A representative piece of the offering was burnt on the fire of the altar, but the rest was eaten by the kohanim.
There are also offerings of undiluted wine, referred to as nesekh.
Parah Adumah: The Red Heifer
The ritual of the red heifer (in Hebrew, parah adumah) is part of one of the most mysterious rituals described in the Torah. The purpose of this ritual is to purify people from the defilement caused by contact with the dead. The ritual is discussed in Numbers 19. If you find it difficult to understand, don't feel bad; the sages themselves described it as beyond human understanding. What is so interesting about this ritual is that it purifies the impure, but it also renders the pure impure (i.e., everybody who participates in the ritual becomes impure).
It is believed by many that this ritual will be performed by the messiah when he comes, because we have all suffered the defilement of contact with the dead. Thus, the existence of a red heifer is a possible, but not definite, sign of the messiah. If the messiah were coming, there would be a red heifer, but there could be a red heifer without the messiah coming.
Sources: Judaism 101