The Round World version is one of the variants of J. R. R. Tolkien's legendarium, published in the final volumes of The History of Middle-earth. In this version, the Earth was always round, and Arda was the name for the whole solar system instead of just the Earth.
In the Round World version the Sun and the Moon were not the fruit of the Two Trees, but actually preceded the creation of the Trees. Instead, the Trees preserved the light of the Sun before it was tainted by Melkor when he ravished Arien.
While this version emerged in the late stage of Tolkien's legendarium, he never entirely decided which version - Flat or Round - to choose as 'real', and while Tolkien was eventually more inclined to use the Round World version, it was never as complete as the Flat World version, and so the latter was chosen by Christopher Tolkien for the published Silmarillion.