We don't know that much about the giants' culture and origin, but the World of Ice and Fire does give us a little info, mostly about where they lived.
I don't think GRRM has mentioned exactly how they came to be (their origin story), and I cannot find anything relevant in any of the books. Also, the only name for their race we have seen so far is giants. There might be something else they call themselves in their own tongue, but we don't know.
On the chapter Ancient History: The Dawn Age there are quite a few mentions about the ways the giants used to live:
Of the giants in the Dawn Age, little and less can be said, for no one has gathered their tales, their legends, their histories.
Ancient History: The Dawn Age
They certainly didn't have cities in the way we people think of them, because they made no houses.
The giants had no kings and no lords, made no homes save in caverns or beneath tall trees, and they worked neither metal nor fields.
Ancient History: The Dawn Age
In the book, there is also some mentions about the places the giants used to live (south of the wall), but people drove them away. Generally, some maesters believe that before the First Men came, Westeros belonged to the Giants and the Children of the Forest.
Were the First Men truly first?
Most scholars believe they were. Before their coming, it is thought, Westeros belonged to the giants, the children of the forest, and the beasts of the field. But on the Iron Islands, the priests of the Drowned God tell a different tale.
The World of Ice and Fire - The Iron Islands
Some of the places that the giants are mentioned to have lived are:
- the hills in Westerlands (and more specifically the Castely Rock) (mentioned in The World of Ice and Fire - The Westerlands and The Westerlands: Casterly Rock)
- the Reach (The Reach: Garth Greenhand)
- the Red Mountains (The Stormlands: The Coming of the First Men)
There has also been some evidence of giants across the Narrow Sea:
- In Norvos (The Free Cities: Norvos)
- In The Grasslands (* Beyond the Free Cities: The Grasslands*)
- In Ib (Beyond the Free Cities: Ib)
There is also another race of giants, named Jhogwin
Jhogwin were the stone giants of Dothraki legend. They were massive creatures said to have been twice as large as the giants of Westeros. They once inhabited the snowcapped northernmost region of the Bone Mountains in Krazaaj Zasqa or White Mountains in the Dothraki tongue, an area that is known as the Realm of the Jhogwin. They became extinct a thousand years ago, leaving behind their massive bones.
The World of Ice and Fire, The Bones and Beyond.