In Australia a Regiment can be two things. It can be a part of a brigade, or in the case of the Royal Australian Regiment in can be part of several brigades.
For example, I am in the 10/27th Battalion which is part of the Royal South Australia Regiment, which in turn is part of 9 Brigade.
Generally an infantry Regiment will have 2 or 3 battalions. I couldn't answer questions about armoured regiments as that isn't my area.
An Australian reserve brigade generally has the following units; an Infantry Regiment, an artillery battery, a signal squadron, an armoured unit of some sort (light cav etc), a combat service support battalion (transport, medical, ordnance,MP's, electrical & mechanical engineers etc) and an engineer squadron or regiment.
A regular army brigade may be slightly differently composed as they have more specialised roles eg. 1st Brigade is Armour heavy with Abrams tanks, two mechanised infantry battalions, 155mm artillery regiment, cavalry, Army Aviation, and an armoured combat engineer regiment.