The nature versus nurture debate is a key theme in The Strange Case of Dr. Jekyll and Mr. Hyde. The fundamental question that the debate tries to address is whether our characters, as humans, are mostly products of our environments or mostly products of innate, inherent personality traits. In the context of the story, we might wonder whether Dr. Jekyll becomes the monster, Mr. Hyde, because of the society in which he was brought up (or nurtured) or because of some innate tendencies towards evil.
On the nurture side of the debate, Stevenson arguably suggests that Dr. Jekyll creates and becomes Mr. Hyde mostly because of the repressive Victorian society in which he lived. As a respectable Victorian gentleman, Dr. Jekyll felt that he had no release for the natural, human desires and instincts which he felt were always bubbling within him. He was, as a respectable Victorian gentleman, expected to at all times repress any such desires and live a Christian life of quiet moderation and reflection. If Dr. Jekyll had lived in a less repressive, more liberal time, he would perhaps have been able to find some manner of release for the desires and instincts which he felt he was unable to satisfy in Victorian England. Because this was not possible in Victorian England, Dr. Jekyll had to repress these desires. The more he repressed them, however, the more anxious and unfulfilled he felt. His creation of Mr. Hyde was, arguably, his only way of finding a release for his desires. If he had lived in a more liberal, less repressive society, he may not have had to repress these desires as much, and he may not, therefore, have had to create Mr. Hyde at all.
On the other side of the debate, it is possible to argue that Mr. Hyde was merely a physical manifestation of an evil that existed naturally within Dr. Jekyll. Whether he lived in a more liberal society or not, he perhaps would have committed the same evil nonetheless, whether as Mr. Hyde or as Dr. Jekyll. One might also argue that Dr. Jekyll's insatiable thirst for knowledge was an inherent trait of his character and that it was this thirst for knowledge which inevitably led to the horrors and terrors he committed as Mr. Hyde.
I think the truth of what Stevenson was trying to communicate is probably somewhere between the two arguments. I think Stevenson was implying that we all have evil within us and that evil is an innate characteristic of humans. But I also think that he was implying that that evil can be exacerbated by repressive social environments. In other words, both nature and nurture have their parts to play in the lives of every one of us.