We based our summary on the Roman poet Ovid's version of the tale (from Book 10 of The Metamorphoses), but in earlier Greek versions, Persephone, queen of the underworld, is also really into baby Adonis. Persephone and Aphrodite (a.k.a. Venus) argue over who gets to raise the beautiful boy. (Is this creepy to anyone else?)
Eventually, Zeus steps in to broker peace between the two feuding goddesses. The verdict: Adonis has to spend four months out of the year with Persephone in the underworld, four with Aphrodite, and four months alone.
The legend goes that Adonis spends his "four months alone" with Aphrodite as well – that is, until he is killed by a wild boar and ends up with Persephone full time. In some versions, Adonis is actually allowed to come back from the underworld for a few months out of the year to be with Aphrodite (source).
This idea of Adonis being a figure who dies (goes to the underworld) and comes back again seems to be directly inspired by the ancient Mesopotamian myths that the Greeks based their version on. From ancient times, Adonis (under different names) was known as a god of regeneration; his death and rebirth represented the coming and going of vegetation over the course of the year.
Both Adonis and Aphrodite came to Greece (via the Phoenicians) from the Middle East. These two divine lovers had lots of different names over the years. Astarte and Adon, Inanna and Dumuzzi, Ishtar and Tummuz, all were names for similar deities. Some say that the Egyptian deities Isis and Osiris were also derived from the same roots. The myths surrounding these two lovers had many different spins over the years. Check out a few here and here.