The Civil War greatly improved the economy of the North but harmed the economy of the South.
The Civil War disrupted the Southern economy badly. First of all, it freed the slaves, thus removing the bulk of the Southern work force and forcing the South to readjust its economy. Second, it devastated much of the South's infrastructure. Destruction of things like railroad tracks caused short-term damage to the South's economy.
Meanwhile, the Northern economy was helped by the war. It experienced a railroad boom as railroads were built with federal government support. This was the most important change in the Northern economy as it led to westward expansion and to a much better transportation grid for the North's economy.
As the response above states, the aftermath of the Civil War left the North in relatively good condition economically and the South in shambles. There were many reasons for this, some because of the respective economies pre-war, some because of where the war was fought, and some because the North won and the South lost.
As pohnpei397 notes, the South lost its free labor, which of course had a devastating effect on the economy. But furthermore, the Southern economy was almost exclusively agricultural, thus far more labor-intensive than an industrial economy, and it also lacked the sort of infrastructure that would have allowed it to diversify and move into a more industrial economy. Additionally, the previous unpaid labor, slaves, were spectacularly uneducated, as a matter of law and/or custom, since literate and educated slaves would have presented problems to their masters. This meant that even skilled paid labor that could have led to industrial diversification was not present in the region.
The North, on the other hand, had a largely industrial economy, with more "universal" education and a rapidly developing infrastructure that allowed it to move produced goods more cheaply and easily. Thus the North entered the post-war years in better economic shape.
Since most of the Civil War was fought in the South, parts of it were ravaged, by occupying forces who did not make any particular effort to conserve resources, by burning, and by other forms of destruction. After all, it was a war. So, in addition to losing its free labor, the South had to do a significant amount of rebuilding.
The North, of course, saw few battles, and even those did not result in nearly the amount of destruction the South had sustained, so it did not need to concern itself with much in the way of rebuilding. All it had to do was use the existing factories, a relatively skilled labor force, and infrastructure to convert from creating wartime products to creating peacetime products.
The South had issued its own currency when it broke off from the nation, and its currency was rendered worthless because it lost the war. Translated into modern terms, this would have been a loss of billions of dollars, I think, and I would guess that the credit terms for borrowing from the North to rebuild would have been very steep, as a matter of risk and as a means of punishment.
Since the North won, its currency prevailed, so it did not diminish in value and was the coin of the realm. Thus, all the wealth held in the North, to the degree people had capital after a long and expensive war, maintained its value.
Today, if one looks at the economies of the North and South, the consequences of the Civil War are still apparent. Southern states have higher rates of poverty, poorer educational outcomes, higher infant mortality rates, and lower life expectancy rates, all of the statistics one would expect when when a region has experienced what the South experienced.