Sacrifices were an important part of life in ancient Greece, not just because of the religious connotations, but because for many, it was the only time one had the chance to eat meat. Although fish was a staple of the Greek diet, and smaller communities could hunt small game, having red meat was extremely rare for any, even the rich.
Sacrifices were modeled as being for the gods; however, the gods got a very small portion of the meat of the animal sacrificed, the rest often going to the people who showed up to the festival, rite, or other occasion for the sacrifice.
This practice began, they said, when Prometheus tricked the gods into taking a portion of meat that was only thigh bones wrapped in fat with a bit of meat upon them. After this, Athenians copied this as tradition, and the rest of the meat of the animal(s) sacrificed was distributed according to the custom of the particular occasion.
However, this was not the procedure followed for all sacrifices. For example, when sacrificing to Hades, a burnt offering was usually placed in the ground, burned completely, and left for the god of the underworld. This was called a holocaust.