This is a frightening tale about the dual nature of mankind. Each of us has a Mr. Hyde "hiding" inside of us, fighting for our souls. There is a spiritual battle between good and evil, not only outside of us, but inside of us and if we are not careful, the evil will take over, just as Mr. Hyde does.
Dr. Jekyll at first can control when he turns into Mr. Hyde, but as he becomes more and more prideful about what his science has created (the potion), he soon realizes that his ugly and evil side starts popping up whenever he feels like it. This story shows how easy it is to slip into things that wind up taking control of our lives if we are not careful. If we give in to our sinful natures, then that nature takes over. With Mr. Hyde, Stevenson shows not only how scary and ugly that evil nature can be but also what it is capable of doing -- all sorts of horrible acts.
In my opinion, Stevenson's story is telling us that the average human being has two sides -- one side that is good and one that is evil. The story is telling us that these sides coexist within each human being. Finally, I think that the story is telling us that the evil side is the stronger and is likely to defeat the good side over time.
In the story, Jekyll can easily become Hyde, but he has a harder and harder time changing back into Jekyll, for example. This shows the power of the evil side. He is also unable to resist turning himself into Hyde -- once again, the evil is stronger than the good. In a way, the evil is even more attractive to the person -- Hyde feels younger and more free than Jekyll.
Overall, then, I feel that Stevenson is arguing that human beings are part good and part evil but that it is easy for the evil to win.