For the iconic period of ten years, immortal Titans from Mt. Othrys waged war against immortal gods of Mt. Olympus (plus two of the Titans, Themis and Prometheus — who would later regret having helped the god who ordered his torture) to determine who would rule the world/universe. This war is called the Titanomachy or War of the Titans, following a tradition that names the war after the loser (e.g., Persian Wars, Trojan War).
For the Titan gods and as many as sprang from Cronos had long been fighting together in stubborn war with heart-grieving toil, the lordly Titans from high Othyrs, but the gods, givers of good, whom rich-haired Rhea bare in union with Cronos, from Olympus. So they, with bitter wrath, were fighting continually with one another at that time for ten full years, and the hard strife had no close or end for either side, and the issue of the war hung evenly balanced. Theogony (11)
Only when the gods had made allies of the hundred-handers (Briareus, Cottus and Gyes, sons of Ouranos) and the Cyclops, whom Kronos had imprisoned in Erebus, could the gods achieve victory.
 But when first their father was vexed in his heart with Obriareus and Cottus and Gyes, he bound them in cruel bonds, because he was jealous of their exceeding manhood and comeliness and great size: and he made them live beneath the wide-pathed earth, where they were afflicted, being set to dwell under the ground, at the end of the earth, at its great borders, in bitter anguish for a long time and with great grief at heart. Theogony (11)
The Cyclops gave the gods valuable gifts in exchange for their release. To Zeus they gave the lightning bolt; to Poseidon, the trident; to Hades, the helmet of invisibility.
Zeus waged the war against Cronus and the Titans.13 They fought for ten years, and Earth prophesied victory14 to Zeus if he should have as allies those who had been hurled down to Tartarus. So he slew their jailoress Campe, and loosed their bonds. And the Cyclopes then gave Zeus thunder and lightning and a thunderbolt,15 and on Pluto they bestowed a helmet and on Poseidon a trident. Armed with these weapons the gods overcame the Titans, shut them up in Tartarus, and appointed the Hundred-handers their guards16….Apollodorus 1
Only after the hundred-handers started to pelt the Titans, could the giant immortals be caught and chained in Tartarus, giving the rule to the gods.
 And amongst the foremost Cottus and Briareos and Gyes insatiate for war raised fierce fighting: three hundred rocks, one upon another, they launched from their strong hands and overshadowed the Titans with their missiles, and buried them beneath the wide-pathed earth, and bound them in bitter chains when they had conquered them by their strength for all their great spirit, as far beneath the earth to Tartarus. For a brazen anvil falling down from heaven nine nights and days would reach the earth upon the tenth: and again, a brazen anvil falling from earth nine nights and days would reach Tartarus upon the tenth. Round it runs a fence of bronze, and night spreads in triple line all about it like a neck-circlet, while above grow the roots of the earth and unfruitful sea. There by the counsel of Zeus who drives the clouds the Titan gods are hidden under misty gloom, in a dank place where are the ends of the huge earth. And they may not go out; for Poseidon fixed gates of bronze upon it, and a wall runs all round it on every side. There Gyes and Cottus and great-souled Obriareus live, trusty warders of Zeus who holds the aegis. Theogony (11)
Atlas, leader of the Titans was singled out for punishment: to hold up heavens upon his shoulders:
After Juno saw that Epaphus, born of a concubine, ruled such a great kingdom, she saw to it that he should be killed while hunting, and encouraged the Titans to drive Jove from the kingdom and restore it to Saturn. When they tried to mount heaven, Jove with the help of Minerva, Apollo, and Diana, cast them headlong into Tartarus. On Atlas, who had been their leader, he put the vault of the sky; even now he is said to hold up the sky on his shoulders.  Hyginus Fabulae
The three main male gods were the three who had received the gifts. Among them, they divided the universe, giving the sea to Poseidon, the Underworld to Hades, and the sky to Zeus.
Edited to add: I just found this quiz. See how much you know about the Titanomachy.