The main infantry weapon was the musket. During the Revolutionary War, the English for some strange reason didn’t want to give us any more Brown Bess muskets (gee I wonder why), so we got our muskets from the French. We copied these to make our own muskets. From about 1795 to 1812 we made low numbers of French style flintlock muskets. When the War of 1812 broke out we couldn’t make them fast enough, so we changed the way we made muskets a lot. By the time we perfected it, the war ended. We continued to make those up until about 1830. Then, we completely changed the way we manufactured them, going to machine stamped parts and such and doing a lot less work by hand labor, but the design of the 1830 musket was pretty much the same as the Model 1812 musket, which was still pretty close to the original French Charleville. So using entirely new methods, we continued to make the same old thing.
In about 1840, things changed. We switched from flintlocks to percussion locks, and then rifled the barrels and started using Minie balls. The Model 1840 was produced as a smoothbore flintlock, but a lot of them ended up with percussion locks fitted onto them before they made it out to the field, and some of them ended up rifled. The Model 1842 was produced with a percussion lock, but with a smoothbore barrel. The 1840 and 1842 were both produced with thicker barrels, anticipating that they would probably be rifled, and many were.
The Model 1840 and 1842 muskets that ended up being rifled were called, unsurprisingly, rifled-muskets, since they had been produced as smooth bore muskets but ended up being rifled later. The term rifle-musket or rifled musket continued to be used for the main infantry muskets, even though they were no longer produced as a smooth bore, as long as the so-called rifled musket was roughly the same length and of similar design to the smooth bore musket it had replaced.
All of these muskets were .69 cal, the same as the French Charleville they were based on. When you go from a round ball to an elongated Minie ball without changing the diameter, you add quite a bit of mass to the round, and this extra mass wasn’t necessary, and meant that they were throwing a lot more lead into each round of ammunition, which meant they could produce fewer rounds for the same amount of lead. So, the next step was to reduce the caliber from .69 down to .58. They also played around with the Maynard Primer system, which is basically strips of paper primer kinda like toy cap pistols use. The Model 1855 musket had the Maynard Primer and the .58 caliber barrel.
The Maynard Primer proved to be an unreliable piece of crap. It claimed to be waterproof, but it’s number one problem was that it wasn’t. For the Model 1861, they went back to percussion caps but kept the .58 cal. barrel.
When the Civil War broke out, they had a big shortage of weapons, so everything back to the Revolutionary War smoothbore muskets was dragged out of the closet. If it could shoot, they used it. A group of soldiers might have a couple Model 1861 or 1855 type muskets, maybe a Model 1840 or 1842, and two or three Model 1812 to 1830 style muskets. When they went into battle, they dropped the old smoothbores and picked up newer percussion rifle-muskets off of fallen soldiers when they could. Meanwhile, the north cranked out as many percussion rifle-muskets as they could, while the south bought most of theirs from foreign countries. The south had some production capability, but not much compared to the north, though they did manage to capture one armory and took most of its equipment back to Virginia.
Cavalry and navy soldiers used shorter carbines that were often based on the same designs as the longer muskets. Some breech loading rifles and later some repeating rifles were placed into service. All of these were produced in much smaller numbers than the main infantry muskets.
Pistols, when used, were usually cap and ball type revolvers.
Flintlocks, like the Hall rifle you linked to, as well as many other rifles and muskets, were often converted to percussion locks in the 1840s and 1850s.
The two most widely produced weapons during the Civil War were the Springfield musket (Model 1861/1863) and the British Model 1855 Enfield musket, which was imported by both the North and South.
In much lesser numbers, you had the Lorenz rifle (often a piece of junk), and other weapons like the Spencer and Henry repeaters. And you had some interesting variants like the Brunswick rifle and Whitworth rifle, which fired specialized rounds made just for them.