OK, good lord this answer stretched on, apologies for my total lack of brevity.
In many ways the American Civil War was the first war of the industrial age, and most of the major innovations spring from that. I will focus on the three most important innovations and the fallout from those, in order of least to most significant.
Steam powered warships and Ironclads
Both of these existed and were used in limited ways in warfare before the Civil War, but the Civil War saw the mist widespread use up to that point. Steam power allowed smaller, heavier ships to move with a kind of speed that would normally require a decent spread of sail, and it allowed them maneuver in any direction without the need of the wind. Armor plating allowed gunships to duke it out with coastal batteries on a much more even footing. Not to mention, the introduction of exploding shells made some kind of armor plating almost necessary on naval warships. All of these things changed the naval landscape considerably, making naval warfare more dynamic and giving it greater capabilities to support land forces. Innovations begun during the civil war (like turreted guns), would radically change the design and construction of warships in the decades going forward, while the need for coaling stations to supply steam boilers would create a rush for colonization in different parts of the world.
The widespread use of rifles
What many people don't realize is that the United States, at the outbreak of the Civil War, was the most industrialized nation on Earth. American factories invented interchangeable parts, and American armories were the first to produce mechanically complicated guns that could be issued to large numbers of soldiers thanks to the ability to obtain and interchange replacement parts. While rifles had been around before the Civil War, factory-made barrels, interchangeable parts, and the minié ball made them feasible to issue in large numbers for the first time ever.
Long range infantry rifles are the biggest contributor to the changes in warfare between Napoleon and WWI. These changes continued throughout the century, but the Civil War is one of the first large conflicts in which rifles were widely used.
To understand why it's significant, imagine standing on the end of a football field with an enemy soldier standing on the other. That's actually about 20 yards longer than the average engagement range during Napoleon's time. If the person was 50-100 yards further, you wouldn't even bother shooting at them, because your musket was that inaccurate. Cavalry charges, bayonet charges, lining up in big blocks, and direct-fire artillery were all possible and all effective because the actual range of a musket was so short. When you expand that range to around 400 yards, everything changes. Frontal cavalry charges are suicidal when every enemy soldier can get off several accurate shots instead of one wildly inaccurate one. Artillery needs to move farther back and shot longer, arcing shots. Massed formations and bayonet charges are similarly useless when the enemy can shoot 5-6 rounds at you while you advance instead of 1-2. Other answers have pointed out the change in tactics that occurred during the Civil War, which is all true. But tactics don't exist in a vacuum, they change because the battlefield itself has changed, and in the case of the Civil War, nearly all those changes were in response to the widespread use of long-range, highly accurate rifles.
The introduction of rifles in the Civil War set all of these things in motion. Cartridges and breech loading rifles, introduced near the end of the war, would push those trends even farther. By the time WWI rolls around, cavalry is nearly useless, standing in the open at any range is dangerous, massed infantry charges are incredibly costly, and almost all artillery has moved way behind the lines.
Railroads were far and away the most significant development in warfare since the invention of gunpowder. Since the beginning of recorded history, every general has faced two hard limits to what he and his army could do. First, 15 miles a day was the average distance an army could travel from the time of Alexander up to Napoleon. Second, 60,000-80,000 men was the largest army you could supply under average conditions; any larger than that and the supplies it consumed in one day would exceed the supplies it received. Also, keep in mind that armies were supplied by animals: mules, horses, oxen, etc. The more men you have at the front line, the more animals you need to supply them. The more animals you have going to the front, the more food you need to supply them. Food for animals at the front lines requires animals in the rear to bring it up, and the whole thing dominoes outward. It's easy to see how supply becomes incredibly difficult when all you have are wagons and mules to bring it up.
When you look at the most successful armies in history, most of them found a way to cheat one or both of those rules by a little bit. Napoleon typically managed 18 miles a day and through clever maneuvering could amass an army of 200,000+ on the day of battle. The Mongols moved at a staggering 40-50+ miles per day. The list goes on. Within the railroad lies the potential to go far beyond just stretching those rules, you can blow them completely out of the water.
For every ton of "fuel" a wagon of six mules could carry 500 tons of supplies, whereas a 15-car train could carry 5,250. The average train of the era moved at about 15mph, meaning it could travel the same distance in one hour that the average infantryman could move in an entire day on foot, and the soldiers on railcars wouldn't even be tired when they arrived. Railroads made some of the oldest rules of war almost meaningless. Huge armies could be supplied with minimal effort; armies over 100,000 now seemed trivial, armies in the millions now seemed feasible. Long "marches" over hundreds of miles could be done in days instead of weeks.
With railroads, war becomes less like a cat and mouse game between two opposing armies and more like a massive chess game between two opposing nations. The strength of your economy mattered as much as the strength of your army. There was no realistic upper limit to the number of soldiers you can recruit if you have the economy to supply them. The massive carrying capacity of railroads meant large battles can now stretch on for days, weeks, or even months without worrying about supplies. All of this began during the Civil War. To see what the use of railroads does when taken to the extreme, you need only look at WWI. Without railroads, it would have been impossible to supply millions of soldiers for years on end at the front lines of WWI. America in 1861 had far more miles of rail than any country on Earth, and the use of rail lines in the Civil War was a major impetus for European nations to expand their own rail networks. More than anything, I think the use of the railroad in the Civil War affected not only future wars, but the entire course of history along with it.