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A squadron was historically a cavalry sub unit. The term is still used to refer to modern cavalry units but can also be used as a designation for other arms and services. It should not be confused with squad.
United States 
In the modern United States Army, a squadron is an armored cavalry, air cavalry, and/or other reconnaissance unit whose organizational role parallels that of a battalion and is commanded by a lieutenant colonel.
Prior to the revisions in the US Army structure in the 1880s, US cavalry regiments were divided into companies, and the battalion was an administrative designation used only in garrison. The reorganizations converted companies to troops and battalions to squadrons, and made squadrons tactical formations as well as administrative ones.
In the British Army and many other Commonwealth armies, a squadron is the counterpart of an infantry company or artillery battery. A squadron is a sub-unit of a battalion-sized formation (usually a regiment), and is usually made up of two or more troops.
The designation is also used for company-sized units in the Special Air Service, Special Reconnaissance Regiment, Honourable Artillery Company, Royal Engineers, Royal Corps of Signals, Royal Army Medical Corps, Royal Marine Commandos and Royal Logistic Corps, and formerly in the now defunct Royal Corps of Transport.
Squadrons are commonly designated using letters or numbers (e.g. No. 1 Squadron or A Squadron). In some British Army units it is a tradition for squadrons to also be named after an important historical battle in which the regiment has taken part. In some special cases, squadrons can also be named after a unique honour which has been bestowed on the unit.
An escadron (the French word for squadron) is another word for a cavalry division. For a long time, an escadron corresponded to a battalion, uniting several companies. Since the mid 20th century, an escadron has been the equivalent of a single squadron (typically 13-tank strong).
In the cavalry (now called the "mounted arm") a captain (3 galons, or braids) commands an escadron (what would be a "company" in the infantry) and is thus called a chef d'escadron (with escadron in the singular). However, his superior in the hierarchy (4 galons) commands 2 escadrons and is thus called chef d'escadrons (with escadron in the plural). There are 2 exceptions – in the Gendarmerie and Artillerie (both accounted mounted arms), such a commander (again with 4 galons) is a chef d'escadron (singular).
In the Swedish cavalry a "skvadron" means a unit with the same size as a company in the rest of the army (about a hundred men). Even Jäger and MP units may have squadrons.
The Norwegian army operates with units called eskadroner (pl.), typically a company-equivalent unit, generally in armoured cavalry units although not always.
The 2nd Battalion, Brigade Nord, has a company-equivalent unit called kavalerieskadronen, or "the cavalry squadron". It serves as the main reconnaissance unit in the battalion. Like the mechanized infantry units, it wears the distinct khaki-coloured beret of the battalion instead of the normal black for cavalry units.
The Armoured Battalion (Panserbataljonen) has the majority of its constituents labeled eskadroner. Including the Cavalry Squadron, the Armoured Squadron and the Assault Squadrons. It also includes the battalion's Support element, the Combat Support Squadron. Its members are also referred to as dragoons, reflecting the nature of the unit.
The Telemark Battalion also has a number of units labelled eskadroner. This includes the Armoured Squadron, the Cavalry Squadron and the Combat Support Squadron.