The nature of a sacrificial ritual as well as that which was to be sacrificed could vary somewhat, but the most basic sacrifice was that of an animal - usually a steer, pig, or goat (with the choice depending partially upon cost and scale, but even more upon what animals were most favored by which god). In contrast to Jewish tradition, the ancient Greeks did not regard the pig as unclean. It was, in fact, the preferred animal for making sacrifices at rituals of purification.
Typically the animal to be sacrificed was domesticated rather than wild game (except in the case of Artemis, the huntress goddess who preferred game). It would be cleaned, dressed up in ribbons, and taken in a procession to the temple. Altars were almost always outside in front of the temple rather than inside where the cult statue of the god was located. There it would be placed on (or beside, in the case of larger animals) the altar and some water and barley seeds would be poured on it.
The barley seeds were thrown by those not responsible for the killing of the animal, thus ensuring their direct participation rather than mere observer status. The pouring of water on the head forced the animal to "nod" in agreement to the sacrifice. It was important that the sacrifice not be treated as an act of violence; instead, it must be an act in which everyone was a willing participant: mortals, immortals, and animals.
Then the person performing the ritual would pull out a knife (machaira) that had been hidden in the barley and quickly slit the animal's throat, allowing the blood to drain into a special receptacle. The entrails, especially the liver, would then be extracted and examined to see whether the gods accepted this sacrifice. If so, then the ritual could proceed.
At this point the sacrificial ritual would become a feast for gods and humans alike. The animal would be cooked over open flames on the altar and the pieces distributed. To the gods went the long bones with some fat and spices (and sometimes wine) - those would continue to be burned so that the smoke would rise up to the gods and goddesses above. Sometimes the smoke would be "read" for omens. To the humans went the meat and other tastier parts of the animal - indeed, it was normal for the ancient Greeks to only eat meat during a sacrificial ritual.
Everything had to be eaten there in that area rather than taken home and it had to be eaten within a certain amount of time, usually by evening. This was a communal affair - not only were all of the members of the community there, eating together and bonding socially, but it was believed that the gods were participating directly as well. A crucial point worth keeping in mind here is that the Greeks did none of this while prostrating themselves on the ground as was the case in other ancient cultures. Instead, the Greeks worshipped their gods while standing up - not quite as equals, but more equal and more similar than one normally encounters.