These games were held at Delphi, in honour of the god Apollo’s defeat of the giant snake, Pytho. As such it, like other sites of pagan worship, attracted the wrath of Christians like Clement of Alexandria, who attacks its foundation story:
Let us now proceed briefly to review the contests, and let us put an end to these solemn gathering at tombs – the Isthmian, Nemean, Pythian, and, above all, the Olympian games. At the Pythian games they worship the Pythian serpent, and the gathering held in honour of this snake is entitled Pythian. At the Isthmus the sea cast up a miserable carcass, and the Isthmian Games are lamentations for Melicertes. At Nemea another, a child Archemoros, lies buried, and it is the celebrations held at the grave of this child that are called by the name Nemean. And Pisa – mark it, you Panhellenic peoples! – your Pisa is the tomb of a Phrygian charioteer, and the libations poured out for Pelops, which constitute the Olympian festivities, are appropriated by the Zeus of Phidias.
Clement of Alexandria, Exhortation to the Greeks 1
These games are notable in that they had both artistic and athletic competitions, which fits with Apollo’s identity as the god of music and the arts; they also seem to have had some sort of arrangement with the Olympics, because the person who won the flute-playing competition at Delphi played the flute during the long-jump at the Olympics.
As for the contests at Delphi, there was one in early times between singers to the cithara, who sang a hymn in honour of Apollo; it was instituted by the Delphians. But after the Crisaean war, in the time of Eurylochus, the Amphictyons instituted equestrian and gymnastic contests in which the prize was a crown, and called them Pythian Games. And to the competition between the cithara players they added both flute-players and cithara players who played without singing, who were to render a certain melody which is called the Pythian Nome.
Strabo, Geography 9.3.10
Plutarch, who was a priest at Delphi, reports a discussion about getting rid of newer additions to the Pythian games and gives us a good account of the various events that took place there. One interesting upshot of adding artistic events was that women could and did compete directly with men in these:
At the Pythian games there was a discussion about taking away all the sports which had recently crept in and were not there from ancient times. For after they had added contests in tragic plays in addition to the three ancient music competitions, which were as old as the games themselves (the Pythian flute-player, the lyre player, and the singer to the lyre), as if a large gate were opened, they could not keep out an infinite crowd of plays and musical entertainments of all sorts that rushed in after him. Which indeed made no unpleasant variety, and increased attendance, but impaired the gravity and organization of the games. Besides it must create a great deal of trouble to the judges, and considerable dissatisfaction to very many, since only a few could win. It was agreed upon that the orators and poets should be got rid of – this decision did not come from any hatred of learning, but as these contenders are the most noted and worthiest men of all, they admired them, and were troubled that, when they must judge every one deserving, they could not give the prize equally to all. I, as I was there at this discussion, dissuaded those who were for removing things from their present settled order and who thought this variety as unsuitable to the games as many strings and many notes to an instrument. When the same subject was talked about at supper, as Petraeus the president and director of the sports, was entertaining us, I defended music, and maintained that poetry was no upstart intruder, but that had been admitted into the sacred games in the dim and distant past, and crowns were given to the best performer. Some guests imagined that I intended to produce some old musty stories, like the funeral games of Oeolycus the Thessalian or of Amphidamas the Chalcidean, at which they say Homer and Hesiod contended for the prize. But passing by these instances as the common theme of every grammarian, as we shall the criticisms of those who, in the description of Patroclus’ funeral games in Homer, read ϱ̔ήμονες, orators, and not ϱ̔’ ἥμονες, darters, as if Achilles had proposed a prize for the best speaker — omitting all these, I said that Acastus at his father Pelias’ funeral set a prize for contending poets, and Sibylla won it. At this, a great many demanded some authority for this unlikely and incredible story, I happily produced Acesander, who has this story in his description of Africa; but I must confess this is not an easy to find book. But Polemo the Athenian’s Commentary of the Treasures of Delphi, which I suppose most of you have read through and through as he is a very learned man and careful about Greek antiquities. In him you shall find that in the Sicyonian treasure there was a golden book dedicated to the god, with this inscription: “Aristomache, the poetess of Erythraea, dedicated this after she had won the prize at the Isthmian Games.” Nor is there any reason, I continued, why we should so admire and reverence the Olympic Games, as if, like Fate, they were unalterable, and never admitted any change since the first institution. For the Pythian Games, it is true, has had three or four musical prizes added; but all the exercises of the body were for the most part the same from the beginning. But in the Olympian games everything except running is a late addition. They added some and abolished them again; such were the apene, either rode or in a chariot, as likewise the crown appointed for boys that were victorious in the pentathlon. And, in short, a thousand things in those games are mere novelties.
Plutarch, Moralia 674d-675B
 The great statue of Zeus at Olympia was sculpted by Phidias (you can still see the remains of his workshop): it was one of the seven wonders of the ancient world.
 c. 595 BCE.
 A mule car race: it was eventually dropped because it was felt to be undignified. The boys pentathlon only lasted for one Olympics before it was dropped.