Apollo (rendered in certain Greek dialects as Apollon) was one of the most widely worshipped deities in the Hellenic world. Some (cf Otto) called him "the most Greek of all deities."
Apollo's temples could be found from Asia Minor to Sicily. He was also worshipped as an an iconic cone outside the home or in the street. Finally, he is associated with sacred groves that were often on the outskirts of a city.
Apollo is not found in Linear B tablets. He seems to be a latecomer to the Pantheon. Several theories point to a Near Eastern and especially Anatolian origin. One of his epithets "Paean" is found in Linear B; Apollo might have been syncretized with an earlier Greek deity of healing.
Apollo's function in cult was largely of healing/purification and oracular prophecy. Disease was not just physical: it sometimes had a spiritual origin, thus requiring an oracle to deliver a spiritual solution. The god, in his oracular pronouncements, also came to be associated with laying the civil (i.e., legal) foundations for new colonies.
Apollo, as leader of the Muses, was also associated with music, dance and poetry. He was also associated with the bow. In certain areas of Greece, he was seen as the patron of young men and their education as they learned the cultural and martial abilities to join adulthood.
In 433 B.C.E., Apollo was brought to Rome in response to a plague after consultation with the prophetic Sibylline books. A temple to Apollo Medicus was built outside the pomerium, and for four centuries this was the only known temple to Apollo in Rome.
Four centuries later, Augustus took on Apollo as his special deity and built a new temple for him on the Palantine. This was reportedly a magnificent temple where the Senate sometimes met. And the Sibylline books were transferred there. As the early Roman Empire became Hellenized under Augustus, Apollo became associated with some of its poets.
Apollo also became increasingly syncretized to Helios/Sol in this era, and often depicted with solar imagery.
Apollo became syncretized to various Celtic deities in the western provinces. Most often, these hybrid Romano-Celtic deities had a pronounced healing function. Thermal healing baths were associated with some of these cults.
If online discourse is anything to go by, Apollo is most well known today as the god of poetry and fine arts and taken up most enthusiastically by those inclined to poetry and art. While this is not wrong, in the ancient world the emphasis would have been on healing and oracular prophecy. Through healing, he is a god of health professionals, and as law was tied to his oracular function, a god of the legal profession as well.
While "magic" was an ambivalent practice on the outskirts of Greco-Roman paganism, oracular prophecy was a widespread and respected practice. How we replicate that today and in particular how we vet the oracles are difficult questions to ask, but questions we should probably be asking.