American Armored Divisions were organized roughly along the same lines as infantry divisions, with a few less combat troops than a "line" infantry division. The numbers and exact organizational structure varies between the North Africa campaigns, the Italian campaign, and the Continental campaigns. Im more read up on the Normandy/France structure, but just know that it didnt vary that much between 1942 and 1944.

Roughly, armored divisions had between 200-300 tanks in them, which includes things from combat tanks (the M4 Sherman) to artillery tanks (the M7 priest), engineering vehicles (of which there were few in World War Two), and other armored support vehicles. They were organized much like infantry divisions, in that each division was made up of platoons, companies, battalions, regiments, and brigades. They were combined arms divisions, which meant that infantry, artillery, and tanks all operated inside of the same unit, and infantry and armor (ideally) operated in close coordination. The biggest difference between an infantry division and an armored one was the "Combat Command" concept. Usually the size of a brigade (about 3-5k men), each division had a combat command (CCA and CCB). They were a hodge podge mix of all kinds of units, infantry, armor, and artillery, that were assigned to one commander to accomplish a specific mission. Each unit was flexible, and was constantly tailoring its strength to the mission at hand. I believe (and please correct me if Im wrong here) that CCs would even pull companies and regiments from other infantry divisions to fully tailor their strength.

All this was to achieve the armored division's ultimate mission: exploitation. Unlike today, American and British tanks during World War Two werent designed to fight other tanks. It was accepted that tanks would sometimes be pressed into service in the Anti-Tank roll, but that job was intended for specially designed tank-destroyer units like the M18 Hellcat. Tanks like the M4 Sherman were designed to rush through a break in the enemies lines, then roll on towards operational and strategic objectives like terrain features, cities and political objectives, enemy supply and command centers, and other areas. All the while, these units would attempt to bypass and avoid centers of enemy resistance which couldnt be immediately rolled through. The name of the game was speed, and American armoured divisions excelled at punching though.