With a title like Adonais, you can bet that at least one Greek god is going to show up. Shelley relied pretty heavily on Greek myth, and those tragic, dramatic characters suit his Romantic elegy quite well.
As a representation of Keats, for whom the poem was written, he appears in nearly every stanza. Adonais was loved and desired by all until another god's jealousy led to his death, much like Keats (at least, in Shelley's mind). He serves as metaphor for the beauty and purity of youth.
Next, we meet his mother Urania, also known as Aphrodite, goddess of love. She first appears in stanza II, and the speaker frequently addresses her throughout the poem. As the goddess of love, she is a symbol for those that cared for Keats, and the speaker frequently admonishes them for not coming to his aid in time to save his life.
Then arrives Apollo, god of music, in stanza XXVIII (249). His arrows are used in a simile for criticism; like an arrow stings its prey, so does the speaker believe criticism wounded Keats.
Shelley takes a break from Greek references and moves to Irish mythology in stanza XXX (268). Ierne, an Irish goddess, shows up to symbolize the Irish poet Thomas Moore. (That was nice of her.)
Shelley returns to Greek mythology in stanza XXXI (276), with Actaeon, the hunter. Actaeon fled in fear after spying on the goddess Diana bathe, and she turned him into a stag to be torn apart by dogs. The speaker compares him to a frail stranger who shrinks back at the sight of Keats' body.
Stanza XXXIII (291) sees the arrival of Dionysus, god of partying. He and his followers come to pay their respects to Keats. Shelley uses these characters to pay homage to Keats; the more arrivals at his wake, the better.