I've recently really gotten into reading about colonial/napoleonic/etc history (book suggestions welcome please, Im reading Merchant Kings now) and Europeans have had a lot of cavalry traditions from the medieval knights to hussars and cuirassiers and spahis and so on. In Western films there's the trope of being saved by the cavalry (and horsemanship in general is a big part of the cowboy mythos, so it's kind of an old American value), but I've also seen mention here that one of the reasons the Civil War was so bloody is because Americans were using Napoleonic tactics without the cavalry arm that Napoleon used.
Did America ever have a cavalry with a fancy special name or traditions? It would also be cool to hear some battles American cavalry were important or notable in, so I could look them up.
During the Civil War, at least early on, we see many flamboyant regiments being raised by enthusiastic volunteers. Such as the 3rd New Jersey Hussars, or the 6th Pennsylvania Cavalry. Casualties, and the initial volunteer enlistment only being two (If I recall correctly) years more or less ensured that regiments like the 3rd NJ or the 11th NY ("Zouaves") with their flamboyant and hard to re-supply uniforms would eventually be phased out.
America's cavalry began to come into its own, strategically, during the Civil War and immediately before and afterwards during the Indian and Seminole wars. Really, America had the most modern Cavalry in this time period, compared to its European counterparts, I touch upon this in a recent reply.
Its a different tradition, unique to America's development, but just because they don't have such specialized cavalry doesn't mean they don't have a strong, amazing cavalry tradition! By the time of the Civil War, the advent of rifles, and generally superior equipment, meant that Cavalry had to move beyond simple tactical reconnaissance and shock action. This is when you begin to see truly amazing deep raids by the likes of Sheridan and Stuart. It should be noted that Buford's delaying action, fought almost completely dismounted, after a long march ahead of the Union army, probably saved the first day at Gettysburg. So like I said, a strong tradition, just a very different one to the European experience.
The Civil War had battles that were quite literally duels between two large bodies of cavalry, and many more general battles where Cavalry played prominent roles. I advise you to look into battles such as "Gaine's Mill" (1862) "Brandy Station" (1863), The 3rd Day of Gettysburg and "Farnsworth's Charge" (July 3, 1863). There's also battles centered around large Cavalry raids, such as Selma (1864) and Trevilian Station (1864), largely considered the largest Cavalry action of the Civil War.
Even during the Revolutionary and Wars of 1812, the terrain of North America lent itself more to Cavalry being dressed, fashioned and trained in the tradition of Dragoons and Mounted rifles. Hope this helps.
Edits: Battles
The absence of Confederate cavalry during the Battle of Gettysburg is notable, as it deprived Lee of his eyes and ears for a battle that developed very dynamically early on.
In that vein, you should read up on J. E. B. Stuart, too, certainly a larger than life character, for better or worse.
Theodore Roosevelt's Rough Riders were as idiosyncratic as its commander.