@Merry Andrew,
Army organization has long varied, and will probably continue to do so as the times and the means of making war change. From the 1780s reorganization until 1871, the French without a doubt had the best organized army in the world, and the Americans copied that in preference to the English system. The French, however, used the "tripod" system. Three divisions (what we would call companies) formed a battalion, three battalions formed a regiment--but brigades were
ad hoc formations. Divisions were formed of as many regiments, troops of cavalry and artillery batteries as the Corps commander thought the commander capable of using effectively. The French also had superior tactical and operational doctrine in that period of more than 80 years. The Americans copied this to a point. Formations higher than a regiment for the period up to the Civil War were usually
ad hoc, and were assignments based upon the judgment of the army commander. Zachary Taylor used regimental formations independent but operating in concert, while Winfield Scott formed divisions from the assignment of regiments and sections of artillery (a section, usually two, sometimes three guns, were the organizational sub-units of a battery).
During the American civil war, it was usual to assign five, and sometimes six regiments to a brigade, and two or three brigades to a division. During the Spanish War, the army reverted to regiments as the central organizational unit, assigning regiments to a commander. So, for example, old Joe Wheeler was put in command of the cavalry which was sent to Cuba, and ended by commanding all of the regiments that were put ashore near Santiago de Cuba (then the Spanish colonial capital, on the opposite end of the island from Havana). He used his regiments as the prime maneuver formation in his assault on San Juan, the key to the defense of Santiago.
In the First World War, the Germans and the French used the tripod system, and a division was made up of three regiments, plus organic artillery and engineer units. The British continued to use brigades, and for them, the battalion and the regiment meant entirely different things. A regiment was the administrative unit, and there were always at least two battalions. One battalion would be the cadre of the regiment, which would train newly admitted levies, and one battalion would be sent to serve in the field. Some regiments had more than two battalions, some as many as five--with one at home as the cadre formation, and the rest serving in the field. A British brigade was made up of the battalions assigned, and batteries of the Royal Artillery. The Royal Artillery and the Royal Engineers were administratively independent of the army.
The Americans in the First World War developed a sort of "hybrid" system. They used the tripod system, and to form a brigade, used three regiments. Two brigades then formed a division, with the artillery, engineers and other support units organic to the division. This meant that in 1917 and -18, many American division were as large as or larger than entire German and French corps (formed of more than one division) because those formations had been reduced by casualties and "wastage."
In the Second World War, the army used a complete tripod system for their divisions. Three battalions formed a regiment and three regiments formed a division. Divisional artillery was a regiment, formed of three battalions, each comprised of three batteries. Generally, a battalion of artillery was informally assigned to support each regiment, and often a single battery was assigned to support each battalion. Engineers and other support units were organic to the division. There were exceptions, of course. Airborne divisions usually were comprised of four, and sometimes five regiments. There would be three PIR (parachute infantry regiments) and a glider regiment. The glider regiment would bring in light artillery, mortars and any vehicles the division was prepared to drop. There might also be a fifth "commando" PIR in a division. Armored divisions were organized on an entirely different model. These would be made up of three combat commands--Combat Command A, Combat Command B and Combat Command R. A and B were roughly equivalent to brigades, with organic artillery and mechanized infantry units, and CCR was the reconnaissance unit of the division.
The army also employed RCTs--regimental combat teams. This would be a regiment--often reinforced--with small support units assigned, so many artillery batteries, an engineer company or two, a heavy weapons company. Arguably the most famous was the 442nd RCT. The 442nd regiment was formed of Japanese-American volunteers from California and the west coast, and it was reinforced with the 100th Infantry Battalion, made up of Japanese-Americans from Hawaii. It was one of the most (possibly the most, but i'm not certain of this) decorated regiments in the army. It suffered in excess of 350% casualties during the war, and fought the nasty, dirty, muddy nightmare war in Italy, and then landed in the south of France.
Today, divisions are, supposedly, made up of two brigades, and how they are organized depends on the mission for which they are intended. All support units, including artillery, are organic to the division, and are assigned missions as needed at the discretion of the division commander.
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