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.
Some formations have a standard composition, while others are headquarters that can control a mission-specific number of sub-units. Even when there is not a flexible headquarters, sub-units can be attached to or detached from a regular unit to optimize it for a mission.
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. Combat battalions typically have 800-1,000 soldiers, although specialized noncombat battalions (e.g., medical or intelligence) may be much smaller.
Headquarters staff sections are prefixed "S", such as "S-3" for the operations staff.
Brigade
In general, a brigade is the smallest type of combined armsformation, and generally consists of two or more battalions plus headquarters troops. Various services and nations have traditional terms, such as Royal Marine Commando or Marine Expeditionary Unit or Armored cavalry regiment.Typical strengths range from 3,000 to 5,000 troops.
Headquarters staff sections are prefixed "S", such as "S-2" for the intelligence staff.
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.
This is the lowest level that customarily has a chief of staff. Sections of staff sections are prefixed "G", such as "S-4" for the logistics staff.
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. Under NATO guidelines, a corps has approximately 50,000 soldiers.
Army
An Army is a grouping of, usually, two or more corps, with a strength of 100,000 or more. 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.
An Army is generally commanded by a General or equivalent.
Multi-national force Iraq (MNF-I) is the size of a small field army, although it is not called one because Third United States Army is the official "army" for the regional United States Central Command.
Army Group
Army groups, unlikely to be seen again in combat, are groups of multiple field armies.