The conflict of this story is one of character vs. Nature. Prince Prospero, and every one of the courtiers he brings with him to his sequestered abbey, fear death, though death is natural. They take such great pains to protect themselves from the disease afflicting their community, and they "bid defiance to contagion," seeming to believe that wealth and youth and beauty will keep them safe from it. It is an extremely proud position, but we see their fear in the fact that they will not enter the last of the prince's seven rooms, the room of black and "a deep blood color," and in their response to the ebony clock's chimes: "disconcert and tremulousness and meditation [...]" every hour. Ebony, or black, is symbolic of death, the clock is symbolic of mortality, and the blood color of the windows of the room in which the clock sits is symbolic of the particular death which they are attempting to escape. Ultimately, however, neither they nor we can avert death because it is natural. Death, the agent of Nature, triumphs.
Both the internal and external conflicts of story pit the individual against death. This is an internal conflict because the story depicts the protagonist, Prospero, as fearing death. People naturally have a desire to live, so that they often must struggle with the inevitable fact that they must die. However, this fear of death also becomes an external conflict for Prospero because Poe personifies death, making it a Masked Figure that leads Prospero into the innermost chamber, faces him, so that he dies. Note that Red Death is capitalized throughout the story, which indicates that it has the attributes of a person with intentions and motivations. Enotes has an excellent discussion on this story as an allegory whereby each objects and characters symbolize ideas in such a way that the entire story can be read literally or figuratively, the latter reading providing the more significant meaning.
The main problem in the story is the plague (or red death) that is around. In order to protect against that plague, the Prince sequesters a thousand of his most loyal followers in his castle. The main conflict in the story is between those inside, and red death itself. They try to keep it out, and Red Death tries to work its way in.