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. An example of the superiority of Ironclad vessels can be seen in the initial engagement with the CSS Virginia (emphasis mine):
Five Union ships mounting a total of 219 guns guarded the mouth of the
James River at Hampton Roads: the Minnesota, Roanoke, St. Lawrence,
Congress, and Cumberland. The last three were sailing ships— pride of
the navy in the 1840s but already made obsolescent by steam. The first
two were steam frigates (the Roanoke was disabled by a broken shaft ),
pride of the navy in 1862. But the fighting this day would make them
obsolescent as well.
she came, heading first for the twenty-four-gun Cumberland, sending
several shells into her side before ramming and tearing a seven-foot
hole in her hull that sent her to the bottom. While this was
happening, the Cumberland and Congress fired numerous broadsides at
the Virginia, which “struck and glanced off,” in the words of a
northern observer, “having no more effect than peas from a pop-gun.” 3
This was not quite accurate; before the day was over two of the
Virginia’s guns were knocked out, every fitting on deck and part of
her smokestack were shot away, her ram was wrenched off by the
collision with the Cumberland, two of her crew were killed and several
were wounded. But none of the ninety -eight shots that struck her
penetrated the armor or did any disabling damage.
After sinking the
Cumberland, the Virginia went after the fifty-gun Congress, raking the
helpless vessel with broadsides which started fires that eventually
reached the powder magazine and blew her up. The Minnesota having run
aground in an effort to help her sister ships, the Virginia turned her
attention to this flagship of the fleet that had captured Hatteras
Inlet the previous August. But the Virginia’s deep draft prevented her
from closing with the Minnesota as night came on. The rebels left the
Minnesota and the other ships for the morrow, and called it a day. And
what a day— the worst in the eighty-six-year history of the U. S.
navy. The Virginia sank two proud ships within a few hours— a feat no
other enemy would accomplish until 1941. At least 240 bluejackets had
been killed, including the captain of the Congress—more than the navy
suffered on any other day of the war.
McPherson, James M. (1988-02-25). Battle Cry of Freedom: The Civil War
Era (Oxford History of the United States Book 6) (pp. 375-376). Oxford
University Press. Kindle Edition.
Innovation on Ironclads continued throughout the Civil War and the Union deployed them in significant numbers. Things like turreted guns can be traced back to the original Monitor design. Ironclads were a revolution in naval affairs, even the all powerful Royal Navy recognized the change in the winds signaled by these new ships (emphasis mine):
When the news of the Monitor-Virginia duel reached England, the London
Times commented: “Whereas we had available for immediate purposes one
hundred and forty-nine first-class warships, we have now two, these
two being the Warrior and her sister Ironside [ Britain’s experimental
ironclads]. There is not now a ship in the English navy apart from
these two that it would not be madness to trust to an engagement with
that little Monitor.”
McPherson, James M. (1988-02-25). Battle Cry of Freedom: The Civil War
Era (Oxford History of the United States Book 6) (p. 377). Oxford
University Press. Kindle Edition.
The widespread use of rifles
What many people don't realize is that the United States, at the outbreak of the Civil War, was perhaps the most industrialized nation on Earth. Owing to a lack of labor and an abundance of resources, it was imperative that American companies find ways to raise productivity to take advantage of them. American arms makers pioneered a process by which the individual pieces of a weapon would be made separately by machines constructed solely for that purpose. It was the beginning of what we would now consider 'interchangeable parts' (emphasis mine):
The French arms industry had pioneered interchangeable parts for
muskets as early as the 1780s. But most of those parts had been
fashioned by skilled craftsmen working with hand tools. Their
interchangeability was at best approximate. What was new to European
observers in 1851 was the American technique of making each part by a
special-purpose machine, which could reproduce an endless number of
similar parts within finer tolerances than the most skilled of
craftsmen could achieve. The British named this process “the American
system of manufactures ,” and so it has been known ever since.
McPherson, James M. (1988-02-25). Battle Cry of Freedom: The Civil War
Era (Oxford History of the United States Book 6) (p. 15). Oxford
University Press. Kindle Edition.
The significance of this seemingly minor difference is hard to understate. Before this system all guns still had to be made by experienced gunsmiths, after this they could be made by factory workers with limited training. Not only that but armies in the field could more effectively make repairs, and in the long run it enabled more advanced weapons to be made. You would not engineer a mechanically complicated rifle if the average soldier had no hope of being able to fix it. An example:
a test of ten randomly selected muskets each made in a separate year
from 1844 to 1853 at the Springfield (Massachusetts) armory convinced
British skeptics. A workman disassembled the parts, jumbled them in a
box, and reassembled ten muskets flawlessly.
McPherson, James M. (1988-02-25). Battle Cry of Freedom: The Civil War
Era (Oxford History of the United States Book 6) (p. 16). Oxford
University Press. Kindle Edition.
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 the standard issue weapon of nearly every infantryman. To understand why that hadn't happened earlier I'll refer again to McPherson's book (emphasis mine):
Given the rifle’s greater range and accuracy, why were not all
infantrymen equipped with it? Because a bullet large enough to “take”
the rifling was difficult to ram down the barrel . Riflemen sometimes
had to pound the ramrod down with a mallet. After a rifle had been
fired a few times a residue of powder built up in the grooves and had
to be cleaned out before it could be fired again. Since rapid and
reliable firing was essential in a battle, the rifle was not
practicable for the mass of infantrymen. Until the 1850s, that is.
Although several people contributed to the development of a
practicable military rifle, the main credit belongs to French army
Captain Claude E. Minié and to the American James H. Burton, an
armorer at the Harper’s Ferry Armory. In 1848 Minié perfected a bullet
small enough to be easily rammed down a rifled barrel, with a wooden
plug in the base of the bullet to expand it upon firing to take the
rifling. Such bullets were expensive; Burton developed a cheaper and
better bullet with a deep cavity in the base that filled with gas and
expanded the rim upon firing. This was the famous “minié ball” of
Civil War rifles.
McPherson, James M. (1988-02-25). Battle Cry of Freedom: The Civil War
Era (Oxford History of the United States Book 6) (p. 474). Oxford
University Press. Kindle Edition.
Minie balls were used in the Crimean War as well, meaning the lessons of these new weapons were already somewhat known. What distinguishes the usage in the Civil War is the scope, and the fact that they were given to lower quality soldiers. The amateurish nature of the early American armies was arguably a more ringing endorsement for the design since they still managed to maintain the weapons and were effective with them.
To understand why range is so 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 Civil War did not introduce rifles, but, as with many other things on this list, the war and the technology of the time combined to test and evolve the technology in a way that hasn't happened before. The American System of Manufactures ultimately made possible metal cartridges and the more reliable breech loading rifles introduced near the end of the war. 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 far behind the lines.
Railroads
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.
Conclusion
The underlying trend in everything I've said here is industrialization. America was certainly not the only industrialized nation in the world at the time, but the American economy was in a unique situation that catalyzed many of these developments in warfare.
The abundance of resources and the lack of labor led to a general need to find many different time-saving and productivity-boosting inventions to create the materiel of war. The American System of Manufacturing, for example, enabled production of rifles on the scale needed to supply the armies, and ultimately allowed for more complicated weapon designs to be produced in larger numbers while still being serviceable by the soldier in the field.
The need to rapidly equip an army that was enormously larger than the one that preceded it led to inventions like sizing for clothes and the first machine to automatically stitch soles onto shoes.
As the final step, when all of these things had been produced the railroads provided a way to resupply the front lines on a scale never seen before. This too would trigger strategic changes as the war went on as well as further down the line.
In the Civil War you have an example of a war in which the population, resources and industrial output of two countries was mobilized in a way that hadn't been seen before, and didn't regularly appear again until WWI. Most European warfare of the era was fast and decisive, economically and politically it was not usually a long haul proposition. The American Civil War gave a glimpse into the future of the highly industrialized, resource intensive, drawn out conflict that would become a hallmark of the 20th century.
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