When someone makes a Horcrux, part of their soul is stored in it, and when the Horcrux is destroyed, so is part of their soul.
In the case of Voldemort, he had split his soul into so many parts that he had irreparably damaged it, so when the 2nd to last part of his soul was released, it appeared in limbo, where Harry saw it. J. K. Rowling has said that after Voldemort was killed, that was all that remained of him.
But what about a person that isn't as twisted as Voldemort. If someone creates a single Horcrux, so they have split their soul, but not to the point that Voldemort has, and then the Horcrux is destroyed, what happens when that person dies? Will their incomplete soul be as damaged as Voldemort's, or will they be able to cross over to the afterlife? Or is there a chance they'll be either trapped in limbo or be trapped on this side, as a ghost?