Cold War armies were organized around the division. For the U.S., it looked like this:
Field Army. 50,000 soldiers. (last used in Gulf War)
Corps. 20-45,000 soldiers. Two divisions.
Division. 10-15,000 soldiers. 3 brigades
Brigade/Regiment. 2-5,000 soldiers. 3-5 battalions
Battalion/Squadron. 500-1,000 soldiers. 3-5 companies/battery/troop
Company/Battery/Troop. 60-200 soldiers. 3-4 platoons
Platoon. 20-50 soldiers. 3-4 squads
Squad. 5-10 soldiers.
“Batteries” are artillery, “companies” are armor and infantry, and troops and squadrons are used by cavalry units. A regiment is generally units of the same kind—all tanks, all infantry, all artillery. A brigade is combined arms, so it takes pieces from various regiments and melds them together into a unit that can fight as a self-contained force.