Berdan's Sharpshooters were the first dedicated "snipers" as we think of them today; experienced in fieldcraft, precisely-accurized rifles, high standards of marksmanship, and the use of optics mounted to the rifle along the bore axis. Amazingly, they were pretty much disbanded and forgotten about after the Civil War, and the concept had to be re-invented by the British and Germans during the early days of WW1.
Berdan's standards were fairly high for the day; candidates had to know how to shoot, place 10 shots onto a 24" disc at 200 yards with iron sights, which was no mean feat back then. These guys were used as scouts, to harrass Confederate supply trains and advance guards, and in general just anywhere a few really good riflemen were needed.
Going back even further, the 60th and 95th Rifle Regiments of the British Army were used to great effect during the Napoleonic wars. Intead of a standard-issue Land-pattern musket, they were armed largely with the Baker Rifle, a shorter and more accurate weapon, much slower to load. These guys were used mainly as skirmishers and picked shots; trained to use cover, work in small teams, and choose targets largely of their own initiative. There were also times they'd be used as a static defense force or in-line with the rest of the army, but were just better shots than everyone else. Most definitely considered an elite unit at the time.
Probably the most famous rank-and-filer of the Rifle regiments was Tom Plunket, who managed to kill a French general at 600 yards firing his rifleon his back, braced with his sling and foot. He then killed the aide that ran up to the general, proving to his fellow soldiers that it wasn't a freak shot.
The 95th weren't the first true snipers, per se, but they were one of the first units trained in specialized tactics and marksmanship, and then issued a weapon to capitalize on it. The French had experimented with rifles off and on, and Napoleon thought them slow and wasteful. French skirmishers and light infantry (voltiguers) were issued the standard duty musket of the day.
The first documented rifle scope was made by Morgan James of Utica, NY; as documented by John Chapman in his 1844 book "The Improved American Rifle". That being said improving rifle sights had begun long before hand, and earlier experiments with optical lenses are bound to have taken place.
The first conflict to see widespread use of telescopic sights, at least to my knowledge, was the American Civil War. But even then it is important to note that the skill and training behind modern snipers is the culmination of centuries of shooting skill and fieldcraft - and the sharpshooters of the Civil War are simply another step in that evolution, and shouldn't be considered snipers in the modern sense.