Was swordsmanship important in the American Civil War?

Generally speaking, swords were not much used during the civil war due to lack of training, lack of quality swords and lack of well-trained horses (plus, of course, improvements in firearms), but there were some notable encounters nonetheless. While swords were used by officers to signal, sometimes for ceremonial saluting and (by infantry with low-quality swords) even for chopping wood, their use in combat was not common:

The US Civil War was part of the beginning of the end for the production of swords for any practical combat use. Firearms were beginning to become more advanced, but the sword still had certain advantages.

Both advantages and disadvantages were evident at the Battle of Fairfield (July 1863), as detailed in the American Battlefield Trust lengthy account, Battle of Fairfield: Grumble Jones' Gettysburg Campaign Victory. For the most part, though,

Bladed weapons are useful, but they were no match for the state of the art small arms (revolvers and rifles) being developed at the time.

Note also that, among Union soldiers,

The vast majority of wounds documented during the Civil War were caused by the Minié ball, while the rest were from grapeshot, canister or other exploding shells. Few men were treated for saber or bayonet wounds and even fewer for cannon ball wounds.

Source: Terry Reimer, 'Wounds, Ammunition, and Amputation' (National Museum of Civil War Medicine)

(my emphasis)

The figures from the U.S. Army Surgeon's report on Union wounds conflict with the above on cannon wounds but otherwise paint a similar picture:

• 50.6% (124,921) rifle / musket
• 42.1% (103,829) unidentified shot (rifle, cannon or pistol)
• 5.7% (14,032) cannon
• 1.2% (3,008) pistol / buckshot
• 0.2% (522) sabre
• 0.2% (400) bayonet

Sources: Mark Hughes, 'The New Civil War Handbook' (2009) & American Battlefield Trust, 'Civil War Facts'

Even accounting for possible sampling bias and other problems with the data (as noted by SJuan76 in a comment below), these numbers would require a huge margin of error for swords / sabres to be considered a major source of battlefield wounds compared to firearms.

Nonetheless, many soldiers did carry some form of blade, some initially out of romantic notions of the past, others because blades did, at times, have their uses. Artillerymen, for example,

carried a short sword used mainly for swiping at charging cavalrymen.

More Detailed answer

The American soldier and writer William Gilham, and those before him, asserted that

“Cavalry is seldom called on to use fire-arms,” wrote Gilham in his chapter on how to fight battles. It was his contention, and the contention of every manual writer before him, that cavalry would be held in reserve until an enemy was broken. It would then be sent to charge them down with swinging sabers. Sabers would also be the main weapon relied upon in fights with other cavalry units.

Source: Philip Katcher, 'The Army of Northern Virginia' (2012)

However, in the American Civil War,

Such could not have been further from the truth. In fact, the nature of war had so changed that the mounted arm became not true cavalry in the European sense, but more like dragoons, men mounted to arrive at the fight who then dismounted and went into action on foot with carbines and pistols. The age of the saber-wielding assault on infantry was gone. Fremantle of the Coldstream Guards, who visited the Confederate Army in 1863, noted the Army of Northern Virginia cavalry with a professional’s eye: “I remarked it would be a good thing for them if on this occasion they had cavalry to follow up the broken infantry in the event of their succeeding in beating them. But to my surprise they all spoke of their cavalry as not efficient for that purpose. In fact, [J.E.B.] Stuart’s men, though excellent at making raids, capturing wagons and stores, and cutting off communications, seem to have no idea of charging infantry under any circumstances....The infantry and artillery of this army don’t seem to respect the cavalry very much, and often jeer at them."

Source: Katcher

On the issue of training:

There was, however, a particularly serious obstacle to the effective use of the saber during the war: the lack of training in its use. In 1861, there was little thought given to attempting to raise volunteer mounted forces comparable to regular cavalry. Colonel Francis Lippitt explained this hesitancy. Since it took three years to train such cavalry properly, it seemed that the undoubted expense of raising such units would be wasted. The war would be over, it was assumed, before they could be deployed.

Source: Gervase Phillips, 'Sabre versus Revolver: Mounted Combat in the American Civil War'

Consequently,

The preference...was to raise light cavalry, ‘of a kind requiring comparatively but little time and training,’ to perform the tasks of outpost duties, patrols, escorts, foraging parties, reconnaissance and providing the advance, rear and flank guards to marching armies. They were not, however, generally trained to deliver charges on the battlefield.

Source: Phillips

On equipment and training,

Nor was it possible at the beginning of the war to equip all troopers with sabers. In June 1861, Jubal Early, then a colonel in Lynchburg, Virginia, complained ‘there is no company of [Confederate] cavalry here fully armed....For many Civil War troopers, the saber was an unfamiliar weapon; even if they were issued one they were rarely fully trained in its use.

This was readily apparent in the way the weapon was handled in the field. The original regulation saber issued to Federal troopers was a rather clumsy, long, heavy sword, of a Prussian pattern. This was later replaced by a lighter, curved saber, a more suitable weapon for light cavalry but still difficult to master. In combat, officers who had been taught to fence used the point of the blade to deadly effect, but enlisted men tended to hack and slash at the head or upper body, often wounding the enemy but without killing or incapacitating him.

Source: Phillips

Concerning the quality of horses available,

The generally poor condition of Civil War mounts favored the revolver over the saber in close combat. The cavalryman relying on edged weapons needed his mount to be nimble, fast and strong for he duelled as much with his horse as he did with his blade. When firing a revolver from the saddle, the condition of the horse was far less important.

Source: Phillips

For the south, the lack of mounts became more critical when they lost control of areas important for horse-breeding, namely Missouri, Kentucky, Tennessee and trans-Allegheny Virginia.

However, there were some clashes where swords were not only used but also bested opponents using pistols, and some officers firmly believed in the use of swords:

...there were commanders on both sides whose faith in l’arme blanche (‘the white arm’) was unshakable. Their belief transmitted itself to their men; properly trained and with sufficiently high standards of horse-mastery to keep their mounts in ‘hard condition,’ these regiments demonstrated the continued potency of the saber, even in the face if the revolver. Jeb Stuart was one such practitioner of saber tactics. During his audacious three-day ride around the Army of the Potomac, 12th-15th June1862, von Borcke recalled ‘we were obliged to fight all the way through, charging continually with sabres in hand the hostile cavalry forces which in all haste were dispatched to oppose us…’Across the lines, there were commanders in blue with a similar attitude; at Kelly’s Ford, 17th March 1863, Colonels Alfred Duffié and J. B. McIntosh brought the 1st Rhode Island, 4th Pennsylvania and 6th Ohio into line, drew sabers and charged, driving Rebel cavalry from the field ‘in magnificent style.’

Source: Phillips

Notable is the Battle of Brandy Station in 1863 where sabres were put to serious combat use:

...on June 9 the Confederates were struck by the Union force in a move that totally surprised them. The result was the first true cavalry battle of the war, with both sides fighting mounted and using sabers in a free-wheeling mêlée that went back and forth for hours.

Source: Katcher

Also, a month earlier in 1863 at Bradyville Pike, Tennessee,

Major-General John Palmer charged 80 troopers of the 3rd Georgia Cavalry: ‘we came on them under a quick fire, but they broke when we got within 100 yards. We pursued them a mile, and have 18 prisoners … The enemy, after they reached the wood, rallied and fought well, but they had no sabers, and only inflicted a few slight wounds.’

Source: Phillips