It is most famous for the oracle of Apollo situated there; the most well-known in all of ancient Greece. To tell the future, the oracle (known as the Pythia) would sit on a three-legged stool over a crevice in the earth. The fumes of the earth would come up and put her into a trance. Then, she would say something unintelligible to all, except for the priests who would write the answer down on a piece of paper.
The answers were often very cryptic. For example, when the Croesus, king of Lydia came to Delphi for advice on invading Persia, her answer was, "One mighty empire would be destroyed." Additionally, he asked how long his reign would be, the answer was "But when the Medes/Persians have a mule as king then, tender-footed Lydian, by the stone-strewn Hermus flee and do not stay and be not ashamed to be cowardly".
He could not believe the Medes would have a mule (half donkey and half horse) as king and went to war with them. However, he did not realise that the ruler of the Persians, Cyrus, had a Persian father and a Mede for a mother, and that the Lydian Empire was the one mentioned in the prophecy. He and his empire were destroyed by Persia.
A list of orcular statements can be found here.