Another couple corrections:
A light infantry section is typically comprised of 8 men, whereas a mechanized infantry section has ten (regular section plus driver and the gunner)
In addition, a platoon is not simply three sections, nor is a company simply three platoons. There are headquarters elements, heavier weapons support and other combat service support elements attached to each which will beef up the numbers.
As for the Battalion structure, it's usually 3-4 rifle companies in addition to combat service support company which used to comprise of 1 x mortar platoon, 1x Recce Platoon, 1 x Anti-Armour Platoon, 1 x Pioneer Platoon (things like mortars and pioneers are being phased out so I don't know the current structure). There's also a HQ company which has a Battalion CP, Int Detachment and MP Detachment and a Sigs Platoon. There's a Logistical Company comprising of Battalion stores, payroll, maintenance, med platoon, cooks and the like.
The term regiment is also a tricky one. There are some war strength regiments that go to war with one Battalion, and others which have 10. The Seaforth Highlanders in Scotland during WWII fielded 10 full battalions. Typically, however, a modern, Canadian Infantry regiment will be allotted 1-3 Battalions. Also, a tank regiment and an engineer regiment are not even one full battalion, so don't let the term "regiment" determine number of troops.
A brigade is (ideally) as listed as Guardian said, and can also have such atts and dets as a helicopter squadron (for air mobile/air assault), a Herc squadron (for airborne) and other bits of shiny kit.
After this, typically formations just get larger.
For light infantry units and Airborne assets, the Brigade or Battalion gets smaller and will be missing certain assets, and have other assets added. For example, you can't parachute an entire regiment of main battle tanks (although the Americans and British are working on that), but you will have assets like Pathfinders, and will trade off certain assets in favour of lighter kit (towed-105 artillery rather than a self-propelled M-109 for example).