I alway assumed that everyone in the Army was a part of one of the 10 divisions. However at Fort Hood they have something called III Corps whose people do not appear to be part of any of those divisions. How is that and what is III Corps (or I and II Corps if they exist)? Are there other units that are not part of divisions? What about ROTC Brigades? Corps of Engineers? Finance Corps?

A division is a combat formation made up of smaller units (the exact terminology is different if you’re talking about Army, Marines or other nations’ forces). A corps is (usually, although not always) an ad hoc formation of divisions for particular operations. Some corps are fairly stable, some are set up for particular conflicts. An example might be XVIII Airborne Corps, which includes the 101st Air Assault and 82nd Airborne Divisions.

However, and I’m sure some of the forces’ posters here will provide the details, in the US Army there are far more support than actual fighting personnel in a division or a corps. They may belong to smaller independent units (battalions, regiments, brigades and so on) that are attached temporarily or permanently to larger units, or they may be part of more generic groups of specialists that don’t operate in large units of their own. Military organisation is incredibly flexible, fluid and complex.

There are a lot of units that are not part of any division. Armored cavalry regiments, for example, are directly attached to corps - they are the highest level of maneuvering reconaissance unit, I think.

Mattk is right that army organizations are fluid and complex; I would add arbitrary. I mean, an Army is bigger than a Corps, which is bigger than a Division, right? The multiple U.S. Divisions in Europe are the main combat arms of a Corps and an Army. The single U.S. Division in Korea is the basis of an Army but not a Corps. A Corps is supposed to be the HQ and hi-level support functions of multiple divisions; an Army is just all the units in a region - it apparently has no minimum size. But I’m writing from memory so I may have mucked it up.