A Little History
The word "platoon" is believed to have originated from the 17th century French word peloton, which is derived from pelote meaning ball, and was used to describe a small "ball" or detachment of men. It is likely that the smaller units that later became known as platoons did not emerge until the advent of musketry, the stable gunpowder warfare that lasted for 150 years, from the late 17th century to the middle of the 19th. Before then, the large and mostly unprofessional armies of Europe relied on the regiment. These large formations did not require small unit maneouvres or tactics, formed as they were of largely "unskilled" pikemen or similar, as exemplified by the confused battles of the Thirty Years War.
However, with the development of the musket and bayonet as well as, and arguably most importantly, drill, small-unit tactics became of extreme importance on the battlefield. Primary among concerns was of not wounding one's comrades when wielding a musket in the firing line, hence the development of specific drill, and by the nineteenth century the concept of volley fire. Armies also became increasingly professional at this time, for example heralding the use of uniforms which helped extinguish individuality along with drill, and one outcome of this was to shrink the size of the forces that went to war.
Platoons therefore arose out of this ongoing development of warfare, as a small formation that enabled a significant amount of firepower to be layed down on the enemy and also allowed easy direction by a newly specialised officer class on the battlefield. The British redcoats would come to master the formation of the platoon most effectively during this time, developing the tactic of firing in "platoon volleys", enabling a near constant barrage of musketry and proving devastating against Indian armies and French columns around the turn of the 18th/19th centuries.
Platoons continued to form an integral part of modern armed forces from then on, through the colonial domination of the British in the 19th century to the bloody battles of the First World War, and from the battlefields of Spain to the fight for beaches of Pacific islands ten years later. By this time the platoon had evolved into something recognisable by today's standards and has undergone relatively little transformation since its inception.
The Modern Formation
Today, platoons are the smallest military units to be commanded by a commissioned officer, in most cases the lowest rank of these, a Second or First Lieutenant. They are designed to be able to perform autonomously on the battlefield and a such have within them light support weaponry, but usually operate within the larger formation of a Company with the added option of Support Platoons and thus the increased firepower and effectiveness that this brings.
In the British Army, the platoon forms one of the basic building blocks of the infantry battalion formation. The infantry can be assigned different roles, for example they can be Mechanised, Armoured or in the Light Role, and the types of platoon found in a battalion changes with the assigned role. Also, the battalion will have three "normal" Companies within it (eg rifle companies in the Light Role) and one "Manouevre Support" Company, which contains the support weapons such as 81mm mortars, MILAN missiles and GPMGs in the Sustained Fire role.
The basic format for all companies, however, remains the same: three platoons to a company, plus HQ. However, since the introduction of "Assault Pioneers", one "normal" platoon from the third company in each battalion has been "lost" and transferred to the Manouevre Support Company, becoming the pioneers. Therefore one of the three companies only has two platoons instead of the usual three. The only difference in this between various infantry battalions is in whether the soldiers are mounted in vehicles (Armoured = Warrior, Mechanised = Saxon) or not (Light Role). The structure of the actual platoon is very similiar: three "sections" to a platoon, with the addition of an HQ and an 51mm mortar team.
"Normal" infantry platoons in the British Army consist of around 40-45 soldiers, and are led by the Platoon Commander, a Second Lieutenant or Lieutenant (pronounced Leftenant), and a Platoon Sergeant, for example a Colour Sergeant. As officers in the British Army change posts every two years or so, the Platoon Sergeant is often relied on to run the administrative side of the platoon when in barracks, and is invaluable to any new Platoon Commander due to his experience (I say his because women are not yet allowed to join British front-line units, i.e. infantry and tank regiments). These two individuals are also charged with pastoral care for the soldiers in that particular platoon.
Platoons are able to operate independently on operations in both peacetime and war, or as part of the larger Company, and so on within larger formations, right up to the Divisional level which would only be used in a large-scale conflict or other operation.
It should be noted that by all means not every unit in the British Army uses the term "Platoon". The exceptions are basically the supporting arms, including the Royal Artillery, the Household Cavalry, the SAS, the Royal Engineers, the Royal Logistics Corps and the Royal Corps of Signals, which instead all use the term "Troop". Troops are, however, identical to platoons in all but name. The Royal Marines, technically part of the Royal Navy but structured as infantry, also use the term Troop, although again the formation is identical to the army's platoon. The equivalent formation in the Royal Air Force and the Army Air Corps is called, appropriately enough, the "flight".
Platoons in the Canadian Armed Forces are largely similar to those of the British, with the Platoon Sergeant acted out by a "Platoon Warrant", as the name implies being a Warrant Officer. Several Canadian Corps also share the trait of using the term troop instead of platoon.
Major Heyman, Charles (retd.): "The British Army: a pocket guide"
Keegan, John: "A History of Warfare"
Wikipedia, damn its eyes.