The prophecy:
When misrule takes its place at the eight corners of the world
When the Brass Tower walks and Time is reshaped
When the thrice-blessed fail and the Red Tower trembles
When the Dragonborn Ruler loses his throne, and the White Tower falls
When the Snow Tower lies sundered, kingless, bleeding
The World-Eater wakes, and the Wheel turns upon the Last Dragonborn.
The first thing I notice is that the events described are an outline of the main chapters of the Elder Scrolls series, and span a period of centuries. The second is that there's a device of implying parallels between the four Towers, the events described are not parallel. In particular, the bit about Snow Tower seems to be talking about the civil war in Skyrim, not about the destruction of the Throat of the World, or to its role in supporting Mundus. The Brass Tower succeeds in the tasks to which it was set. By contrast, Red Tower did more than tremble.
So with White Tower, I think this refers to the Altmer Dominion's sack of the Imperial City. But the city was retaken, and the physical Tower remains intact.
What do the Stones and Towers do? Somehow the Towers harvest creatia and the Stones focus it. Brass Tower changes history. Red Tower was the basis for the Tribunal, who were protectors of a land and a people that was, don't forget, prosperous and stable for millenia. We're still working out Snow Tower, but I believe it has to do with the refuge of heroes, Sovngarde.
White Tower maintains the barrier that keeps Mundus distinct from Oblivion. I think this be obvious from the plot of TES IV. We know it's still working because Mundus has not merged with Oblivion.