Question

How did colonial powers govern their colonies?

Answer 1

I am going to assume here you are referring to the 'Scramble of Africa' that happened in the second half of the 19th century, as the European power did not really control the African regions before then. 

The methods contexts did differ per colonising power and colonised region, but it boils down to the following factors:

- superior firepower, equipment and recourses; having better guns, armour, communication technology, and supply routes, made the Europeans a formidable enemy that the various tribes simply could not counter.

- co-opting the local elites; a tried and tested method for centuries, this has always been the way smart conquerers could maintain control over a region with minimal fuss and expenditur.

- divide and conquer; conflict between the many tribes of Africa has been a constant for centuries in the continent. The Europeans could easily manipulate the various tribes against each other to prevent a unified resistance from rising up. 

- a willingness to use extreme forms of terror; the Europeans might have been all high and mighty back home about their Enlightment and democracy, but in Africa they were more than willing to use forms of terror that would make most contemporary dictators feel a little uneasy. Case in point, the widespread killing and mutilation when quotas were not met in king Leopold II's Congo.

Answer 2

They governed by direct or inderect rule