Pantheistic Religion: Many gods, all with the form of human beings.

Immortal, feasted on nectar and ambrosia, and their veins flowed with ichor. They dwelt on Mt. Olympus in northern Greece.

Each god was associated with certain aspects of human existence, agriculture, and the arts, and was often a patron (like a modern-day Catholic Saint) who could be propitiated with sacrifices and other gifts.

The Twelve Olympians (from Wikipedia)

The gods were constantly getting involved in human affairs. They were typically jealous, competitive, and cruel, but could also give glory to a particular human being whom they favored. For example, in the Birds, the character of the Hoopoe was a human, Tereus, who was transformed into a bird by the gods after violating his wife's sister and cutting out her tongue (people were pretty nasty in mythological times). Also, Apollo and Artemis killed Niobe's fourteen children because Niobe boasted she was a better mother than their mother Leto. Apollo defeated the satyr Marsyas in a musical contest and flayed him alive.

The Children of Niobe

Marsyas

The Greeks sacrified to their gods in great civic festivals. There was no separation of church and state as in modern times. This is important in the Birds with respect to the smoke that rises from the burning meat. The smell of the smoke was supposedly pleasing to the gods (they didn't eat it, after all). Sacrifices took place within peoples' homes, at temples, and during festivals such as the Great Dionysia and the Panathenaea. Goats, roosters, and cattle were typically the animal victims.

Greek sacrifice

The Greek gods were powerful but not omnipotent as many people now conceive of the god of the monotheistic religions. They could be tricked, bargained with, bribed, and even punished. Zeus threw Hephaistos from Olympus and many of the older generation of gods (the Titans) were imprisoned because they fought against Zeus and the Olympians. Prometheus, the Titan god who makes an appearance in the Birds, was punished for tricking Zeus and giving fire to humans. He was chained to a rock and every day a bird ate his liver, which then, being immortal, grew back. In the Birds he merely avoids being seen by the gods.

Prometheus

Humans could interact with the gods as outlined above, but at the end of the day they were usually powerless against them. When mortals were especially saucy to their immortal masters, or, in other words, when they exhibited hubris, they were thought to be struck with nemesis, or divine revenge. So, people who tried to get the better of the gods, like Pisthetairos in the Birds, were usually destroyed or their lives made miserable. The Birds being comedy, however, everyone gets along in the end. In a tragedy the results were usually not as cheery.

The Greeks were aware that other civilizations ("barbarians") had their own gods. The "Triballian" god in the Birds is one example. Sometimes other cultures' gods were assimilated into Greek civic religion. Egyptian gods are a good example.