A squadron was historically a cavalry sub unit. It is still used to refer to modern cavalry units but can also be used as a designation for other arms and services.
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. 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 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 company (typicall 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).