At the beginning of the war, there were over 525,000 outdated Model 1816 and 1840 muskets left over from previous conflicts in United States armories. Most of these had been converted from the old flintlock to the newer percussion firing mechanism, and some were even rifled; smoothbore weapons were becoming obsolete. Southerners quickly seized whatever weapons they could from those arsenals within their states.
The small arm that the Union armies most used is the Model 1861 Springfield Rifled Musket, usually .58. Over 1 million Springfields were made during the war. This featured a rifled barrel that allowed a projectile to spin as it left the muzzle, thus making it more accurate at many more yards. The projectile most used with these was the Minie ball, which despite its name is shaped more like a modern bullet. It was invented in 1848 by a French officer, Claude Minie. What made the Minie ball significant was that its base would expand as it left the muzzle, which caused its spin, and made it able to kill at up to half a mile.
Another popular weapon among the Union armies was the Sharps Models 1859 and 1863 rifle, forerunners of the modern sniper rifle. The Army bought 12,000 of these for the 1st and 2nd U.S. Sharpshooters, who became some of the most renowned units of the war.
The Union cavalry favored breechloading carbines during the war, which look sort of like shorter versions of rifles. The most popular of these was the breechloading Spencer carbine, which was lever-activated and magazine-fed. The Cavalry first received these in October 1863, and by war's end, over 95,000 had been made. The Sharps carbine was the most popular single-shot carbine, which used .52 cartridges and were very reliable. There were many other models of carbines, and unfortunately, the ammunition size for them was never standardized during the war.
As to handguns - which were generally used by infantry and artillery officers and cavalrymen as well as some privately purchased by enlisted men - the most popular calibers and models was the .36 Colt Navy and .44 Colt Army revolvers (1860). Up to 80% of the Union cavalry is thought to have used the Colt Army revolver at some point during the war. Second to the popularity of the Colt was the Remington, which furnished Old and New Model .36 and .44 revolvers. The Remington was similar to the Colt, only less expensive and easier to change cylinders.
The Confederacy started the war with no ordnance department of its own. So it had to start from scratch. There were plenty of manufacturers willing to make a buck making equipment and weapons for the Confederacy, but many of them were woefully inexperienced and there was a lack of standardization among weapons and types of ammunition used. As mentioned, it procured whatever weapons it could from Union arsenals; 296,000 firearms were captured, many of which had to be converted to the newer percussion system. Also, right before the war, Southern states were buying whatever weapons they could from Northern and European manufacturers for their state militias. Often, Confederate soldiers used whatever guns they captured from the Union on the battlefield.
In 1861, the Confederate Chief of Ordnance Josiah Gorgas sent agents to Europe to buy more modern firearms for their army. Many of the European imports were leftovers from the Crimean War of the 1850's. The most popularly used small arm among the Confederate troops was the Enfield, imported from England (and was also popular with Union troops). Over 400,000 Model 1853 Enfields were used during the course of the war. It was a .57 caliber, which meant it could also use the .58 Minie ball.
Over 100,000 Lorenz rifles were also imported from Austria. As the blockade slowly choked off Southern ports, it became more difficult to import weapons - and eventually, much of anything else Southern soldiers and civilians used. 
All told, only 15% of Southern weapons used in the field were Confederate-made. The two most productive armories were those in Richmond and Fayetteville (NC). In addition to these, were was an almost infinite number of private and state manufacturers. Still, European imports and Union captures did the most to arm the Confederacy.
Most carbines used the by Confederate cavalry were captured from the Union armies. Many pistols and revolvers had also been holdovers from state militias and from Union arsenals, many of which needed to be converted from flintlocks. Still more had been purchased in the north just prior to the war's beginning. Colt and Remington revolvers were popular among Confederate officers, too. They also relied on European imports, particularly the Le Mat from Belgium, England, and France. A much smaller number were made by the local ordnance industry, but those working within it were mostly unskilled.
So where do women tie into this? They were most of the ones rolling and filling the cartridges as well as making other accoutrements in arsenals. Women and girls were favored for this kind of work because their smaller hands made them more dexterous and efficient at rolling cartridges. They also took up less space and could be paid less than men. Most cartridges used during this time were paper, and when they and the powder inside got wet in the field, they were rendered useless.
Loading and firing a musket or rifle during this time period was a time-consuming process; an experienced soldier could get off about 3 shots per minute. A soldier had to tear the cartridge with their teeth (indeed, having one's two front teeth was a requirement for service), pour the powder down the muzzle, drop the bullet down inside, drop the cartridge paper in after it, ram it all down, then prime the gun and put a percussion cap over the nipple (which would ignite the powder when the hammer struck it), and *then* ready, aim, and fire. Lather, rinse, repeat. The newer breechloading repeating rifles and carbines were therefore a godsend to those soldiers that received them.
Images from: http://www.civilwarweapons.net
Information from William C. Davis, The Civil War Reenactors' Encyclopedia. Guildford, CT: The Lyons Press, 2002.
*Parrott Rifle - An example of the newer rifled artillery very popular on both sides. Made of iron instead of the traditional bronze. Had band in rear for reinforcement. Weighed less than the Napoleon but twice as accurate. 
* 3inch Ordnance Rifle Model 1861 - Probably most popular piece of the war. Also known as Rodman rifles. Could use same ammo as the Parrott due to them both having the same sized larger bores. 
*Whitworth - a 12-pounder breechloading gun developed in England, which saw some use especially in the Confederate armies, including at Gettysburg. It had a hexagonal barrel and could only fire its own oddly-shaped projectiles, but up to 5 miles. 