I will speak about the United States because that is the area i am most familiar with.

There were several justifications coming from several angles. The biggest argument was economic. The economy of the south was dependent on large resource producing industries which were based on slave labour. Furthermore, any negative impact on the resource producing Southern economy would negatively impact the resource dependent production economy of the North. This meant that despite teh fact that slavery was illegal in the North many northerners still saw it as a necessity and even though many found slavery distasteful arguments for its abolition were slow in gaining traction.

A second group of arguments were based upon the idea that slavery was beneficial to the slaves themselves. These arguments were usually based in pseudoscience or problematic theological interpretation. The idea that white people were more evolutionarily advanced than Africans allowed for arguments that it was right that they should be enslaved. In some cases this line of argument was softened by saying that enslavement in the "civlised" world was better for Africans than freedom in savage Africa. Several fairly tenuous appeals to the Bible also supported slavery, these essentially arguing that the African race was cursed by God to serve the White race.

The scientific and "Biblical" arguments were stonewalled by counter arguments by abolitionist groups and it was the economic necessity which held on longest.