Military formation (ground)
A formation in military terms is a grouping of units. An alternate military use is a temporary topological relationship among units, such as echelon or wedge.
Battalion
A battalion is typically a grouping of companies, usually three of a basic type (e.g., infantry), a support or heavy weapons company, and a headquarters.
Brigade
In general, a brigade is the smallest type of combined armsformation, and generally consists of two or more battalions plus headquarters troops. Typical strengths range from 3,000 to 5,000 troops.
Various services and nations have variant terms, such as Royal Marine Commando or Marine Expeditionary Unit.
Division
Military divisions date back to the Napoleonic era. By the time of World War I, a division generally consisted of two or more regiments/brigades and numbered between 10 and 20,000 soldiers. It is usually commanded by a major general.
Corps
A corps is a unit of two or more divisions; usage of this term dates back to before the American Civil War. Corps are usually commanded by lieutenant generals, although corps that are not independent of a field army may be commanded by a major general.
In some armies, a corps is geographic rather than tactical, or may represent a branch of specialization.
Army
An Army is a grouping of, usually, two or more corps. Armies may also be responsible for the zone of communications behind a major front. They can also be repositories for "army-level" assets, such as super-heavy artillery, heavy armor, or other units of a specialized type not appropriately controlled by lower level headquarters.
A Western Army is generally commanded by a General or equivalent, although there are special cases, often for historical reasons. Eighth United States Army in Korea is actually a reinforced division attached to a larger headquarters organization.
In the Second World War, Japanese "armies" were equivalent to Allied field armies; the Japanese equivalent to a field army was an "area army".
Army Group
Army groups, unlikely to be seen again in combat, are groups of multiple field armies.