I know people have suggested the possibility of slavery being made illegal in the constitution, but here's a different angle on it. What if it had been done in a way that allowed for a gradual fading out of slavery. I.e. in the constitution it would state that every child born on United States soil is born a free and natural citizen of the United States. Children born to slave woman would then be free born children. They would technically be under the rule of their mother's owner as minors, but once they reached adulthood (age 21), they would be free to live their lives however they wished.
Some states would certainly protest, but I think at that point the country would still be new enough and slavery not so deeply entrenched that its abolition would be strongly protested, especially since slave owners would not immediately lose all their slaves.
Slavery really took root in the South because of the invention of the cotton gin in 1793. By the time of the civil war, cotton was the big money crop in the South, made possible by the combination of slave labor and the cotton gin. So before the cotton gin came along, the abolition of slavery would not have been such a hard blow to slave owners.
The states might have protested the law, but several founding fathers, including Patrick Henry, would have been in favor of it. With enough good orators speaking in favor of the law, they could have swayed public opinion toward it.
So, if such a law had been passed, how would things be different? Slavery would be a much more distant part of our nation's history, with the majority of the last slaves having passed away before 1900. Would the Civil War happen at all? How would race relations change? Would the civil rights movement happen sooner, perhaps around the same time as the women's right's movement?