Army organization during the Civil War followed a system developed after the Mexican War. Many sources refer to regiments, brigades, divisions, and other military units. The following is meant to give you an idea of what these units are.
Up to 100 men (though usually fewer) under the command of a captain assisted by two lieutenants. Identified by letters A through K (omitting J, to avoid confusion with I).
The basic infantry unit, made up of ten companies. Typically 400 to 800 men, depending on recruiting, casualties, and sickness. Commanded by a colonel, with a major serving as executive officer.
Individual states, through their adjutant generals, were responsible for raising regiments and, to some extent, for equipping them. After receiving some basic training and drill instruction, a regiment would be mustered into federal service, where it was designated by state name and numbered by seniority; thus, for example, the First Minnesota Infantry.
In the early years of the war, prominent citizens took on the expense of recruiting companies and were rewarded by being elected captain of the group. The states assembled companies into regiments and appointed officers to command them. This system led to considerable political string-pulling, and both sides had many officers with little formal training. Some proved adept at soldiering, but the majority were more interested in furthering their careers. They caused many headaches for commanding officers in the armies of both the North and the South.
In the Union Army, a group of, typically, four regiments under the command of a brigadier general. The regiments often came from different states and could be shifted from one brigade to another as needed; it was not unusual for one regiment to serve as part of several brigades during its career. A brigade typically was known by the name of its commanding officer. The Iron Brigade and the Irish Brigade were two of the distinguished brigades that earned notable nicknames.
Usually made up of four brigades, commanded by a brigadier general or a major general and known by the name of its commander.
Corps (sometimes known as a "grand division")
Several types of units grouped together-three or more infantry divisions and several batteries of artillery, for example. Designated by a roman numeral or by the name of the commanding major general.
Two or more corps plus unattached units of artillery and cavalry. Armies on both sides were known by geographic names, often the names of rivers (the Army of the Potomac, the Army of the Cumberland). Some had regional designations (the Army of Northern Virginia).
Another way of grouping the combatants is by infantry (soldiers on foot), cavalry (soldiers on horseback), and artillery (soldiers assigned to cannons). These groups formed units as follows:
The basic field or light artillery unit commanded by a captain and comprised of four to six cannons, 25 to 30 men to a cannon. Batteries were divided into two-cannon sections, each under the command of a lieutenant. Each cannon was under the command of a sergeant. Horses pulled the cannons into battle on caissons; depending on the size of the battery and the number and size of the guns, there would be anywhere from 45 to 98 horses.
Heavy artillery (also known as foot artillery or siege artillery)
Originally intended to defend fortifications and operate big siege guns. Organized into companies and regiments along the same lines as the infantry. Sometimes, as in the Army of the Potomac, pressed into duty as infantry.
Organized into regiments of from 10 to 12 companies referred to as troops. As the war progressed, cavalry formations began to operate in divisions and later in corps augmented with artillery.
Groups of cavalry and infantry too few to make up a regiment. In some cases (the First Minnesota Veteran Volunteers, for example), a battalion was essentially a reorganized regiment that had suffered so much attrition that it could no longer act as a regiment. Under the recruitment system of the time, the states preferred to form new regiments rather than furnish replacement troops for the "old" ones.