Variants, not one version
Haudenosaunee tradition tells this story in more than one way, and this retelling follows the shape recorded by Arthur C. Parker (Seneca) and J.N.B. Hewitt (Tuscarora) from named Seneca, Onondaga, and Mohawk tellers, without claiming any single version is the only correct one.
The fall
In the Sky World above, a great tree that lit the sky is uprooted, tearing open a hole. A woman, curious or pushed, depending on the teller, falls through it toward the waters far below. In Parker's Seneca summary the chief of the Sky World, jealous and influenced by the being called the Fire Beast, uproots the tree, and as she falls she clutches seeds and earth from the edge of the hole, which later sprout on the new land. Geese or swans, seeing her fall, fly up together and slow her descent, carrying her gently down rather than letting her strike the water.
The turtle's back
The animals of the water world below debate what to do; several offer to dive to the bottom of the endless sea to bring up mud for her to stand on, and many fail before one, muskrat in many tellings, succeeds, bringing up a pawful of earth. The Great Turtle offers his back to bear this earth, and as it is spread across his shell, it grows outward into land, this is Turtle Island, the name still used for the continent today.
What follows
Sky Woman's daughter is born on this new earth, and her grandsons, the twins Sapling and Flint, are born after her, go on to shape the earth's features in their turn. Their story is told separately on this site as its own entry.
After the Twins
In Parker's Seneca summary, Sky Woman, the twins' grandmother, eventually returns to the Sky World, and the good-minded twin later makes the sun and moon from his mother's body; other tellers give those deeds to Sky Woman herself. Tellers also differ over whether Sky Woman sided with Flint against Sapling. The land built on the turtle's back remains, and the name Turtle Island is used today by many Indigenous nations for North America.