The Oracle at Delphi was the most popular and respected Oracles of the ancient world. It was associated with the god Apollo, who defeated the Python, which had been the guardian deity of Delphi. At Delphi, located on the side of Mt. Parnassus where the stone Cronus mistook for Zeus landed, a priestess over the age of 50, called a Pythia, was placed upon a tripod. Later writers state that this tripod was located over a fissure, from which fumes would rise. After she was asked a questions, the Pythia would then achieve an ecstatic state and produce the oracle, which must have been chaotic at best. The priests would then interpret these ramblings for the petitioners, often into poetic meter.
Those seeking an oracle would have left an offering, making the more popular oracles and the priests who attended them quite wealthy. Bribery for a more favorable oracle certainly was not out of the question.
Oracles, especially the Oracle at Delphi, had a tremendously powerful impact upon the ancient Greek world. Powerful men, cities and nations would consult them before making any major decision. Often times, oracles appear to have taken on their own political agenda. In the Persian Wars, the Oracle at Delphi clearly favored the Persians over the Greeks. In the Peloponnesian Wars, it favored the Spartans over the Athenians.
Oracles retained their hold upon Greek society until the first century AD. At that time, they seem to have been replaced by astronomy, a medium much more accessible than the oracles.
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Herodotus. The Persian Wars.
Thucydides. The Peloponnesian Wars.