This is probably the most common scenario posted on AH.com, but that doesn't mean it was likely at all. The Confederacy simply had too many advantages in the War of Secession: the were fighting on the defensive, they had superior generals, their cavalry was far superior, the development of the rifled musket had given tremendous tactical power to the defending army, and the Union had to protect long and vulnerable supply lines in order to advance into Confederate territory.
The only plausible way the Union could have won has to be Kentucky not joining the Confederacy. Until General Fremont stupidly advanced his troops into western Kentucky and announced that he was abolishing slavery in the state (a move which horrified Lincoln, by the way), there was the possibility that the Unionists in Kentucky could have kept the state in the Union. But once Kentucky seceded, I think the game was pretty much up. The Confederacy was too strong to be subjugated.
As for what would have happened, it's difficult to say. The precedent of the Confederacy seceding is what allowed Deseret and the Pacific Republic to split off from the United States in later decades. I suppose that all of these could have remained united under the United States in a US-wins scenario.