There are many different ways to answer this question and many different paths to explore to be able to answer this question.
Throughout all of history there have been examples of slavery and the world undoubtedly would have been different had slavery not existed.
In Ancient Egypt, the Jewish people were enslaved by the Pharaohs. Through our historical writings, the bible, and other narratives and stories we understand that the Jews were vital in the building of the pyramids. Is it possible the pyramids would not have been built without them? Maybe. Also, the discussion of the Jewish people in the Bible would have been entirely different. Moses led the people out of Egypt because they were enslaved; had they not been enslaved, they would not have had this desire. This of course is not the only slavery throughout our history
Throughout all of history we have seen instances of slavery. The most poignant example in America is that to which I believe this question refers, African American slavery. Clearly the history of our country would have been vastly different had there not been slavery. Our economy was greatly assisted by having slaves. Our laws, especially as related to land owning, home ownership, and voting were written with the opposition of slaves in mind. Schools did not allow black students to learn with white students. A black person could not be free in the South. African Americans as a whole were banned from public places, could not drink from the same water fountains, attend the same schools (and at times not even attend school), and have many of the rights and privileges we take for granted. African Americans were beaten and killed, simply for having a different skin pigment. To say the least, the history of the African American people was clearly altered by history. Had there not been slavery in America, our country's history and that of other countries would have been different as well.
In both examples, our world would be greatly different if there had been no slavery. Both groups of people I discussed had to change the way they led their lives or risk death. A new culture even developed because of the way in which slaves were forced to live. Whether our world would have been better or worse, we cannot know; what we do know is that history and the lives of these peoples has been altered.