That second link in the previous post is particularly helpful, especially if you read all three pages of it. Scroll to the bottom and either click pages 2 and 3 in order, or click View All so you can read the entire article. It's an easily-digestible read that hits the basics without getting into the more complicated aspects of the situation. Though it doesn't cite sources (Torah, Neviim, K'tuvim, Mishna, Gemara, Zohar, Shulchan Aruch, or any of the rest of the vast array of Jewish canon and commentaries), it is a very good overview of the mystical aspects of sacrifice and why we can substitute prayer in the time when the location of the Temple Mount is currently in other hands.
Mind, that article pretty much covers only the spiritual reality. In practicality, the sacrifices had a concrete function as well.
There are several types of sacrifice, mind you -- meal, oil, money, incense, prayer, "wave" (of sheaves or loaves) -- and only one type, the burnt offering, used animals as the vehicle, and that was only the offering for the "accidental wrongdoing which one subsequently discovers and wishes to repent."
But all of the sacrifices had/have their uses. The money, for instance, was collected as tithes and offerings, and used to support the widow, the orphan, and the outcast. Meal, oil, and animal meat were "burnt" (which is to say, cooked, not actually burned to charcoal), then eaten. Some parts of some sacrifices were eaten by those who brought them, but were shared by Levites and Koheins, while some were eaten solely by the members of the hereditary priestly castes: Levites, who sang and played instruments at Temple and who worked to maintain the Temple and grounds, and also Kohanim, who perform the higher priestly duties to help the community maintain its relationship with the spiritual and the holy.
See, Leviim (and Kohanim, who are a subset of Leviim; and their wives and children and household members) are the one tribe that were not assigned an ancestral territory as the others were. Since they had no land, they couldn't raise crops, couldn't raise livestock, essentially couldn't feed themselves at all. They can only eat (while the Temple is standing) if the other tribes brought sacrifices and gave offerings to sustain them. They lived by tzedakah (which is often translated as 'charity' but actually means 'justice' and I think gives you a good indication of how seriously we take our obligation to give it).
So if I sinned by mistake, and realize it or discover it, then I would immediately make provision to bring an appropriate animal to the Temple (or provide the money for an animal to be bought and brought). I would give the animal to the Levites who were working at the Temple that day. The animal itself would be slaughtered in humane fashion while I watched, an essential part of the proceedings, because it would immediately throw me into a state of hyper-awareness: THIS is the consequence of my unthinking action. The animal's blood would be drained for a manner of days so that it would be kosher, then it would be butchered to remove the kosher parts of the animal from the non-kosher parts. Non-kosher parts were then given/sold to the non-Jewish community, the "stranger who lives in your gates."
The kosher parts of the animal were then brought to the Temple, where I would be waiting to see my personal sacrifice "burnt" (cooked, roasted) on the altar. The essence and fragrance would be for Hashem. While it cooked, and the essence and fragrance rose, I would spend the time praying and sincerely repenting of my wrongdoing. I'd be thinking, "How can I right the wrong I have committed? Who was inconvenienced, hurt, or G*D forbid killed because of what I did without realizing what I was doing? And what can I do to be more mindful of my actions, so that another animal doesn't have to be brought any time soon?" A Levite or Kohein would sit with me and listen as I prayed the prayer of "bowing down the head," helping me if I didn't know the prayer, helping me to truly understand what I'd done, why I'd done it, how I could prevent myself from doing it again in future.
From the moment I realized my accidental wrongdoing, to the moment when I selected the animal for slaughter, to the moment when it was killed in front of me, to the moment I returned to the Temple and saw the sacrifice on the altar, to the moment when I made the blessing and took my bite of the meat -- acknowledging in a concrete way my part in the entire process -- to the moment when the Levite or Kohein took his bite and completed the ceremony... this was all a very intense emotional experience. One went through sensations of burden, sadness, anger, pain, a heightened realization of culpability, and finally the elevating understanding that one does have the power to accept responsibility and change one's actions. It was a very cathartic thing.
It has been suggested by some that the famous notion of "Jewish guilt complex" didn't even exist until the Temple wasn't there to help us work through our responsibility and redirect our spirituality towards moving forward rather than dwelling on past wrongdoings.
But, thank G*D, we do have prayer. It is called "service of the heart," and while it does require a lot more emotional work (because there are no handy 'props' to play it all out before our eyes and
show us what we need to grasp), it is less expensive, and has the benefit of being able to be done everywhere in the world, rather than only at the Temple Mount.
For (yet) more information, try the
Jews For Judaism post on this subject. In fact, spend a lot of time there, and you'll get answers to LOTS of your questions about Judaism, then and now.
