A brigadier general will generally command a brigade, although the rule does not seem to be strict.
In the Australian Imperial Force during World War I, the rank of Brigadier General was always temporary and held only while the officer was posted to a particular task, typically the command of a brigade. When posted elsewhere, the rank would be relinquished and the former rank resumed. This policy prevented an accumulation of high-ranking general officers brought about by the relatively high turnover of brigade commanders.
The rank of Brigadier General in the British Army was abolished in 1922 (for its insignia refer to General). It was replaced initially by the rank of Colonel Commandant and then in 1928 by the rank of Brigadier. The old name of Brigadier General has since been reintroduced in Canada and some other Commonwealth countries.
In the Royal Air Force and many other airforces that do not use ranks based on those of the army, the equivalent rank is an Air Commodore.
In the Israeli Defence Forces a Brigadier General is called Tat Aluf and it is the third highest rank. It is followed by Aluf (Major General) and Rav Aluf (Lieutenant General) or General).