What is the meaning of the title Heart of Darkness and how does it relate to the story?

There are a number of ways to consider the meaning of the title on its own or as it relates to the story.  You may or may not be aware that Africa was frequently referred to as the "Dark Continent," and Conrad was almost certainly aware of that. The darkness of the continent could refer to its unknown quality or the darkness of the people who resided there. (Remember that the book takes place during a period of colonialism and great racism, and "darkness" in this context might very well have racist overtones today.) 

Another interpretation of the title that you might want to consider is the darkness of men's souls, which could refer to their lack of morality, the darkness of insanity, or the cruelty of colonialism. Remember also that this is a trip upriver, and the source of a river might be considered its "heart," hence the destination might be the source or heart of the darkness of Africa or the men who colonized it. 

Darkness might refer to the darkness of the insanity that the narrator finds at his destination, the darkness of the treatment that he observes on his way upriver, or the despair of the Africans who are so horribly mistreated. When you consider the plot of the entire book, there is little that could not be reasonably called dark.