A Brigade is a collection of different Regiments and supporting units that have been grouped together for a specific purpose. A fighting Brigade will traditionally contain Infantry, Cavalry and Artillery Regiments together with many supporting cap badges. The composition of each Brigade will differ depending on its responsibility but could often contain 5,000 soldiers. A Division would traditionally be made up of 3 or 4 Brigades depending on the specific role it is to undertake and is configured in a similar fashion to a Brigade but on a larger scale. 1 (UK) Division and 3 (UK) Division are fighting Divisions whereas 2, 4 and 5 Division are responsible for administrative support of specific geographical areas.
The Regiment is often considered to be the most important unit in the British Army. It carries the spirit of the people who have gone before and would usually contain approximately 650 soldiers depending on it's cap badge and role. Sometimes Infantry Regiments have more than one unit of this size and they should be correctly referred to as a Battalion and be numbered in ascending order. An example being the 1st Battalion of The Parachute Regiment which like the 2nd Battalion and the 3rd Battalion contains an identical structure and number of posts.
Corps is a term used to describe a collection of Regiments or small groupings of soldiers that share a common area of specialist expertise. It is an organisation that has been developed to ensure that common practice is generated across all members of the groupings and to ensure that common interests can be catered for efficiently.
additionally
The Household Division is made up of seven regiments and is commanded by The Major General. There are two Regiments of Household Cavalry, the Household Cavalry Regiment and The Household Cavalry Mounted Regiment. Both Regiments have Squadrons of The Life Guards and The Blues and Royals. The other five Regiments are collectively known as the Foot Guards. They are The Grenadier Guards, The Coldstream Guards, The Scots Guards, The Irish Guards and The Welsh Guards. The Regiments owe their origins to the fact they were raised for the express purpose of guarding the Sovereign. A duty they still carry out today. This is reflected directly in the Divisional title. Although it is, in part a Ceremonial Duty, the role is operational in support of the Police. The seven Regiments are unique from the rest of the British Army in that they carry out this task as well as their modern military role.
From the British army website, I'd put the copyright but I can't find the button. I tried the Americans
http://www.army.mil/A-Z.htm but I couldn't find anything useful on a quick glance, maybe you can.
I hope this goes some way to answering your question, even though it is incomplete.