Let me try and field some of these. I was Air Force, but stationed with Army for a few years, and since most of these are Army terms, let me give it a shott.
Platoon: A group of soldiers usually around 50 or so.
Company: Smaller than a Platoon. Soldiers grouped together that have similiar skills and such.
Battalion: On the command level a Battalion is a large group of soldiers comprising many career fields grouped together. Approx. numbers I can't remember.
Brigade: Battalion but bigger.
Division: The highest on the Army's Command list I believe. Meaning that the Army is broken down into divisions. 10th Montain division, 2nd and 3rd infantray divisions (of which I was a part of).
Section: Not sure, but sounds self-explanitory
Squadron: A group in the Air Force that comprise career fields that all help each other. My last squadron was the 10th OSS (Operations Support Squadron). It was comprised of Weather (me), ATC, Flight Ops, Admin, Pilots, ect.
Wing: Where an Air Force "Command" is equal to an Army divison, the Wing in the AF heirachy is just one step lower than a Major Command.
Battle Group: Sounds self-explainitory, but can't be specific
Group: See above.