Sacrifice was the locus of the sacred in the ancient world and still is in many parts of the world today. It is high time to restore these ancient and venerable rites to contemporary Pagan practice.
In the ancient world, like unto elsewhere today, the rite of animal sacrifice was central to the experience of the Gods. We can remember the way of sacrifice by using a sketch of Walter Burkert's reconstruction of the Greek rite. It was a three-phase process following on sundry ablutions and preparations. The participants would process to the altar; purify the space, people, and offerings with water; likewise fumigate them with incense; and inscribe a circle on the ground with the participants inside. Then, the rite proper would begin. Preliminary offerings would be poured out and prayers made, addressing the offerings to the Spirits and Deities intended. Then, further and probably more elaborate and formal invocations were performed, and the animal dedicated would be swiftly slaughtered. The animal theretofore had been especially well treated and kept calm and was by this considered 'willing' to be sacrificed. The animal was immediately butchered and the traditionally prescribed portions placed in the fire on the altar while the rest was cooked or distributed. After this, a third phase of concluding offerings and prayers are made.
Think about this a moment: How different is this from an ordinary Sunday Church Barbecue with exceptionally fresh meat? The victim was killed in a respectful, even holy manner, a far cry from today's factory farms. Certainly, no one who eats meat today can have an ethical objection to this practice. Any of the mechanical considerations of hygiene are simply a matter of skill, lost today in our culture, but restorable by careful consideration or learning from those cultures that still perform this venerable rite.
Some in their ignorance of the ancient world and how offerings are generally done even today associate sacrifice with privation and pain, a God killing His Only Son in expiation, for example. While it is perhaps noble to make offering with your last or most precious bit, sacrifice is not based on suffering. Most sacrifice is done in a mood of thanksgiving and comes from the abundance of the offerer. In the ancient world, after the offering, the rest of the animal generally was cooked and eaten in a mood of celebration. (Dancing was another suppressed dimension of ancient Mediterranean religion: few public rituals did not include it. I'm guessing that is it was what you did while dinner was cooking.)
The purpose of sacrifice is to build, maintain, and correct our connection with the Gods, which is why it had to be stopped in ancient times. It is essential for theistic Pagans, but I know atheist Pagans who join in the practice. The common explanation of sacrifice is to somehow 'feed' the Gods, but this is generally challenged by the more philosophical understandings of ancient religion that evolved over time. In the West, this view is championed by Iamblichus of Chalsis and found in the book we now call De Mysteriis, arguably the cornerstone text of the western magical tradition. Iamblichus points out that the Gods and all the entities down the hierarchy of being are above humans on the ontological scale and so cannot be affected, never mind fed, by such as we. Rather, sacrifice properly done affects the sacrificer by attuning us to the Gods we invoke (never mind bonding us to those we share it with). A careful analysis of Iamblichus' writing discloses that the act of offering physical substances to the Greater Ones is best understood as a kind of material invocation.
Material offerings themselves are part of a continuum of offerings stretching from the material to the purely noetic or 'mental,' with a kind of hybrid between them made of both 'matter' and 'thought'. Speech is an example of this, which is both material (being sound, experienced through the senses) and mental (having meaning, experienced and understood by the mind). This middling sacrifice, between the material and the mental, can be termed 'symbolic.' The attentive practitioner will immediately recognize that symbolic offerings composed of words are otherwise called 'invocations.'
In his Of the Abstention from Animal Food, Porphyry argued against animal and all material sacrifice. He is often cited by Christians for this, which is ironic since he was one of their greatest opponents. (They destroyed his massive work, "Against the Christians," such a loss . . .) Iamblichus objected to Porphyry's thesis by noting that most folks can't do the purely mental offerings Porphyry enjoined, and that material offerings are required for material benefits from the sacrifice. In Iamblichus' formulation, it seems that the benefit derived from a sacrifice comes through with the same degree of materiality as the offering. Mental offerings produce mental benefits and physical offerings produce physical benefits.