Americans, and especially Southerners, understood that slaves made for the cheapest form of hard labor possible. Today, American businesses still use this method, relocating factories to third-world countries where workers make only a tiny fraction of America's minimum wage. It's legal, and American businesses are booming because of this terrible trend.
Chattel slavery as instituted by the European colonial powers had specific and unique influences on modern society. Enslavement and disease wiped out all of the original inhabitants of Hispaniola (now Haiti and the Dominican Republic). North American slavery gave settlers access to free labor for their cash crops. This labor force meant that Americans could compete in the global marketplace with a huge advantage. This advantage eventually led to U.S. dominance of the global economy. The current position of the U.S. as a world superpower is intimately connected with its history of slavery.
One major way in which slavery shaped modern society by helping to make the South (though this is changing to some degree) the poorest area of the country. Slavery caused the South to be essentially a colonial economy that was not able to industrialize and modernize on its own. It caused the South to be an area where much of the populace was woefully undereducated. These sorts of factors are ways in which slavery caused the South to become and remain "backwards." This was a major way in which slavery shaped modern society.
Given the tags you have included, I think you are referring mostly to slavery in the United States, which is a pretty broad topic without even getting into more ancient enslavements. Slavery promoted colonialism in Africa, harming it in ways that are manifest today, a whole other topic, created much of the developing United States' infrastructure, provided economic development, in the South and the North, with free labor, and created what appears to be a perpetual underclass in the United States, to this day. In order to enslave people, you must regard them as subhuman, capitol, rather than people. If you look at plantation inventories, you will see that slaves are listed with monetary value, a sure sign they are not regarded as people. That regard, sadly, seems to have carried through in many ways to the present day, still affecting at least this modern society, economically, socially, medically, and even legally, with many African-Americans coming up short in all those measures, all as a result of slavery.