Divisions were the first regularly constituted military formation that combined infantry, artillery, and cavalry, an innovation of Napoleon Bonaparte. Before the invention of the division, commanders had to make ad hoc formations for every engagement, an especially challenging task since the modern staff had not been invented. 
Napoleonic divisions were preconstituted, but did not have standardized mixtures of troops or equipment. The Duke of Wellington refined Napoleon's idea and created interchangeable divisions of standardized capabilities. Divisions were used in the American Civil War, but without significant standardization.
The Prussians, who were also refining the staff system, brought standarized division and a controlling staff together. The combination, organized by Helmuth von Moltke the Elder was a critical force multiplier for his troops in the Franco-Prussian War of 1871.
Divisions in the World Wars
By the First World War, divisions were quite standardized, although some were sufficiently large to be unwieldy. Depending on the country and branch of service, the chief unit of maneuver within a division was a regiment or brigade. The division also had supporting units such as artillery and engineers, which might be kept under central control or attached to maneuver units.
A popular organization of the time was a "square" division of four maneuver units. Sometimes, four regiments would be split into two brigades, which became the unit of action.
Higher-speed action in the Second World War led to "triangular" divisions of three regiments, where two regiments would assault and one would remain in reserve, waiting for a breach it could penetrate. In some armies, there might be three infantry and a tank regiment, or two infantry and one mechanized/motorized infantry unit. These were cases where the penetration unit was specialized for the purpose; foot-mobile infantry could create a breach for fast units, possibly with close air support, to exploit.
Divisions for breakthrough and penetration
To help force that breach, the Soviet Union formed division-sized artillery formations, called "breakthrough artillery divisions".
Divisions for special operating environments
Several of the major powers in the Second World War organized parachute infantry divisions, also called airborne or paratroop. Given the limited capacity of troop-carrying aircraft, these divisions were very light in equipment, but posed a deeper threat than other formations. They were complemented by slightly heavier air-landing divisions, which expected to land on a newly captured airfield.
Other light divisions specialized in mountain warfare.
Divisions in the Cold War
Pentomic was not the answer
During the Eisenhower Administration, under Maxwell Taylor as Chief of Staff of the Army, the U.S. Army reorganized for the "atomic battlefield" into what they called "pentomic divisions", a portmanteau of "pentagon" and "atomic". The "pentagon" referred to their being built around five maneuver units called "battle groups", bigger than a battalion and smaller than a brigade. Given that a division commander has to command artillery, engineers, logistics, staff, and quite a few other supporting organizations, five maneuver elements, especially with no computer support, was too much for one general to handle.
The Pentomic concept also created a real problem for officer professional development, since it eliminated the brigade level. Battle groups were essentially reinforced battalions, commanded by lieutenant colonels. Removing brigades meant that there was no way for leaders to gain operational experience between the level of battalion and major general battalion command. Available logistics technology was not up to keeping independent battle groups supported. The Army eventually gave up and went back to the triangular formation, modified during the Vietnam War with the addition of an aviation brigade, which was not a conventional maneuver element.
In Vietnam, the basic structure of a division was triangular, with three brigades. When the first airmobile division arrived, it had a fourth brigade, but this principally supported the three infantry brigades.
At the Battle of the Ia Drang, North Vietnam created its first division command, under Chu Huy Man; this was a conscious experiment to learn how to fight at the level of divisions. It was later that the People's Army of Viet Nam learned to operate at the corps level; moving from regiment to division was a large step. It was an especially large step to face a new but advanced division, the 1st Cavalry Division (airmobile) under an experienced commander, Harry Kinnard, which was part of a standing corps-equivalent, I Field Force Vietnam under Stanley Larsen.
Twilight of the Division
The current Restructuring of the United States Army changes from a division-centric model to a model of combat and support brigades. The Unit of Employment (UEx) roughly corresponds to a division headquarters, controlling up to six brigades. A number of functions previously controlled at division level, however, have moved to the brigades.
- Stark, Rodney, The Organizational Age: 19th Century Organizations, Sociology, 3rd Edition