Okay, Artemis and Apollo were twins. They looked alike, acted kind of alike, and had similar interests. They were both the children of Leto, the Goddess of Dark Nights and Zeus, the Supreme Ruler of Gods. I'm sure you've heard that they were born on the island of Delos. Well they weren't. They were born on Ortygia which is near Delos. I'll explain, but first we must start from the beginning.
It really started with Leto. Leto was just chillin' one day, hanging out and doing her divine duties when Zeus comes along looking for a little action. Now I probably don't need to tell you this, but Zeus is one serious hunk and a real charmer, so when he proposed a little hanky-panky Leto didn't object. They started making love, but Gods are kind of kinky, so they decided to transform into quails while they made love.
Now Zeus thought he was pretty slick, and that Hera (his very jealous wife) hadn't found out. But she had. So Hera sends Python (this big 'ol snake-monster thing who happened to be Hera's parthenogenic son) to chase Leto all over creation, and decrees that Leto could not be delivered anywhere on solid ground (or where the sun shines, in other versions). Poor Leto eventually made it to Ortygia, where Artemis was born. Artemis was a really good daughter, because when she was born, she saw her mother in pain, and helped Leto over to the straits between Ortygia and Delos.
There, between an olive-tree and a date palm (the two sacred trees of Leto as the Egyptian and Palestinian Lat), Apollo was born. It was on the North side of Mt. Cynthus on the island of Delos (although many surrounding islands and nations dispute that). He was no easy baby. Despite being only seven months in the womb, it took Leto nine days of labor before she could bare the God of Light.
They say that until Apollo's birth, Delos had been a floating island, but as the baby boy was spilled upon its sand, the island grew roots and had been fixed since.
There are other versions of the birth (there always are). The one above is most widely accepted, but there are other things too. Some say that Apollo was born underground. Homer says he was born in Lycia. The Ephesians say he was born on Ortygia (near Ephesus), and two other random countries claim him as their native son as well. He is much loved, is Apollo. Artemis has more reference in the myth above. Originally (pre-Hellenic myth), Artemis was an orgiastic goddess - and as that, the quail was her sacred bird. The island of Ortygia (Quail Island) was the resting place of flocks of quails as they migrated north in the spring.
That's all for now. If you have any questions, or need any more information, just write to me.