I. The Coming of the Titans
Each of the male Titans joined with one of his sisters to produce children. Kronos married his sister Rhea, but was told by his parents that he would be defeated by his own son. To thwart this prophesy, he swallowed each of his and Rhea's children as they were born - Hestia, Demeter, Hera, Hades, and Poseidon. Being immortal, this did not kill them, but they remained trapped inside him.
Rhea grieved for the loss of her children. So, when she was close to giving birth to Zeus, she consulted with her parents Gaia and Ouranos. They revealed the future to her, showing her how to thwart Kronos. First, Rhea went to the island of Crete to give birth to her son. When he was born, his infant cries were drowned out by the Kouretes, attendants of his mother, who clashed their weapons together. He was kept hidden in a cave and reputedly nursed by a goat named Amaltheia, although in some versions Amaltheia was the owner of the goat. The horn of this goat may have been the famous horn of plenty [term to learn: cornucopia] (a detail added by Ovid, but possibly with precedent).
When Kronos came to Rhea for their child, Rhea gave him instead a stone, wrapped in cloths. Not noticing, he swallowed the stone instead.
The infant Zeus grew quickly - Hesiod's Theogony says it took only a year. Between his strength and the advice of Gaia, Zeus was able to force Kronos to throw up first the stone, and then all his siblings one by one. Alternatively, according to the Apollodoros, the Titaness Metis tricked Kronos into swallowing an emetic.
II. The Titanomachy
What happened immediately after [Kronos regurgitated his children] is not clear, but the war between the gods and Titans - the Titanomachy - soon begins. Unfortunately the epic poem of that name, which would have told us much, is lost. The first complete account we have is in Apollodorus (which was probably written in the 1st century A.D.).
Some of the children of the other Titans - such as Iapetos' son Menoetius - fought alongside their forebears. Others - including Iapetos' other children Prometheus and Epimetheus - did not.
The war was fought without success on either side for ten years (a traditional period for a long war; note that the Trojan War also lasted ten years), with the gods based on Mount Olympus, and the Titans on Mount Othrys. These two mountains flank the area of northern Greece called Thessaly, Olympus to the north, and Othrys to the south.
Since both sides of this war were immortal, no permanent casualties were possible. Finally, however, the gods triumphed with the aid of older powers.
Ouranos had long ago imprisoned the three Cyclopes and the three Hundred-Handers (Hekatoncheires) in dark Tartaros. Again advised by Gaia, Zeus freed these monstrous cousins of the Titans and was rewarded with their aid. The Cyclopes gave lightning and thunder to Zeus to wield as weapons, and in later accounts also created Hades' helmet of darkness and Poseidon's trident.
The Hundred-Handers provided more direct assistance. In the final battle, they kept the Titans under a constant barrage of hundreds of thrown rocks, which together with the other gods' strengths, particularly Zeus' thunderbolts, overcame the Titans. The defeated Titans were hauled down to Tartaros and imprisoned there, and the Hundred-Handers became their jailors.
Or at least that is how Hesiod concludes his vivid description of the battle. However, elsewhere in his Theogony, and in other poems, we see that in fact many of the Titans did not remain there.
The children of Iapetos had varied fates - Menoetius was like his father cast into Tartaros, or destroyed by Zeus' thunderbolt. But the varied fates of Iapetos' other sons - Atlas, Prometheus, and Epimetheus - did not involve imprisonment for fighting in the war.
Many of the female Titans or daughters of the Titans - such as Themis, Mnemosyne, Metis - were also obviously not imprisoned. (Perhaps they did not participate in the fighting.) In any case, they became the mothers of the Muses, Horai, Moirai, and - in a manner of speaking - Athena.
The mythological record is silent on most of the rest of the Titans, but a later myth said that Kronos himself was eventually released by Zeus, and he was assigned to rule over the Isles of the Blessed, where the spirits of heroes went after death.