The Muses
Theogony begins with an invocation of the Muses, meaning that Hesiod calls on the Muses to give him inspiration for his poem. The Muses are nine sisters who live on Mount Helicon and give inspiration to artists and poets when they are invoked.
There are nine Muses, and each is responsible for a certain art or talent: Calliope is the Muse whose domain is epic poetry; Clio is the Muse of history; Euterpe is the Muse of music, song, and lyric poetry; Erato is the Muse of love poetry; Melpomene is the Muse of tragedy; Polyhymnia's domain is hymns; Terpsichore is the Muse of dance; Thalia is the Muse of comedy; and Urania is the Muse whose domain is astronomy.
Gaia
Gaia is the Earth. She is the first primordial goddess, who comes out of the nothingness that was the universe before the gods. She is responsible for giving birth to the mountains, the sea (called Pontos), and mountain nymphs. Gaia eventually gives birth to Uranus, and they become the first ruling couple of the gods. Up until Uranus' birth, all of Gaia's creations had been the result of virgin births — she created them herself.
With Uranus, Gaia gives birth to several of their children. These children include the Cyclopes, hecatoncheires, and the Titans. Kronos is also one of their children, and Gaia's favorite son. As such, when the time comes for Kronos to overthrow Uranus, Gaia aids him by giving Kronos a sickle of adamant to castrate Uranus with, successfully dethroning him.
Uranus
Uranus is the sky/cosmos. He is born from Gaia, and together they have three Cyclopes and three hecatoncheires as children. Uranus hates these children due to their monstrous forms, and locks them in caves for their lives.
Uranus is dethroned by his son Kronos when Kronos uses a sickle of adamant to castrate Uranus. Once castrated, Uranus loses all his power, and Kronos becomes the new supreme god.
Kronos
Kronos is the son of Gaia and Uranus. After dethroning Uranus, Kronos and his wife, Rhea, become the rulers of the gods. Kronos and Rhea have six children together, the youngest of which being Zeus. Kronos, afraid that his children will one day usurp him the same way that he had his father, decides to eat his children to eliminate the threat that they pose to his power. Rhea is upset by this, and tricks Kronos into eating a rock instead of their youngest, son, Zeus.
Unbeknownst to Kronos, Rhea hid Zeus away on Mount Ida until he was an adult. Once he reached adulthood, Zeus confronted Kronos, made him throw up the children that he had eaten, and together they waged war against him. Kronos lost the war against Zeus and his other children, and Zeus became the new king of the gods.
Rhea
Rhea is Kronos' wife who disproved of his plan to eat all of his children so that they could not usurp him. She hid Zeus on Mount Ida so that he could survive and one day overthrow Kronos.
Zeus
Zeus is established as the final king of the gods in Theogony. After dethroning Kronos during the Titanomachy (the battle between Zeus and his siblings and Kronos and the Titans), Zeus and his wife, Hera, take up the mantle as king and queen of the gods. Unlike his father and grandfather before him, Zeus remains king of the gods and is never overthrown and remains the king of the gods to present day.
Aphrodite
Aphrodite is the goddess of love and sexuality. After Uranus is castrated by Kronos, Kronos throws Uranus' genitals into the sea off the coast of Cyprus. Aphrodite is born from his genitals.
Eros
Eros is the primordial god of love and desire. Eros is one of the first beings to be born after Gaia, and is described as a fundamental power and cosmic motivator. Eros later becomes personified as Cupid.