Father Time was the "giant of giants" who once ruled a kingdom in Overland, who slept throughout most of Narnia's history in an enchanted sleep in Underland, and who helped bring about the end of Narnia.
Assumingly, he once ruled a kingdom somewhere in the surface world of Narnia, but at some point, he sank down into Underland and was put to sleep there in a cavern, along with Birds, Bats, Lizards, and Dragons. When Eustace Scrubb, Jill Pole and Puddleglum were on their quest to find Prince Rilian, the Gnomes told them that they believed he dreamed of the going-ons of the overworld.
It is unknown what happened to him when Underland fell apart, although he presumably carried on sleeping.
When the end of the world came, Aslan called "Time" several times, which woke him up. He then helped Aslan end the world: in particular, he destroyed the sun by squeezing it like an orange in his hand, which brought an end to all ages. He was then given a new name and left outside at the end of the world.
He did not enter Aslan's Country, and what happened to him afterward is unknown, but it's presumed that he froze to death and his soul joined the souls of the creatures that helped him destroy Narnia.
The narration says that while he was far bigger than any of the giants, "his face was not like a giant's, but noble and beautiful." He also had a large snowy beard, which covered him to his waist, and it is said that "a pure, silver light rested upon him", though "no one saw where it came from."