As with Stevenson's other short stories, the writing tends to be incredibly dense so as to squeeze a lot into a much shorter space. However this particular story takes about a couple of hours and involves mostly a dialogue between the protagonist (if you wish to call him that) and an unnamed supernatural being whom we are lead to believe is the devil, but we are never specifically told this.
The story takes place in a shop in which the protagonist is looking at purchasing some goods, however something happens and causes him to act in a rather peculiar way. Basically he kills the shop owner and then begins looting the store of all of its valuables. As he is in the process of looting the store he hears movement upstairs, and when he investigates he encounters this mysterious fellow and they begin to converse.
The conversation has a lot to do with the nature of evil and while Markheim attempts to justify his actions, this person debates him at every point resulting in him admitting that he is actually not a good person. Well, that is obvious from the fact that Markheim stabbed the shopkeeper and was planning on robbing him, but at this point he has had an epiphany. Basically he has been confronted with his inner nature, the fact that he is an evil person, and instead of hiding from it he is forced to confront it.
This is something that many of us have to go through, but in another sense many of us will continue to ignore our dark side rather than face up to it and admit that it exists. This has some very Christian connotations, though I would not necessarily suggest that Stevenson was writing a gospel account, or attempting to convince anybody that they are a sinner. However, in the same way that Dr Jeckyl is overcome by Mr Hyde, and his evil nature over rides his good nature, so to we see here with Markheim and the inner struggle that his good side and his evil side are having.
While it is suggested that this is the devil, the problem is that this is not necessarily the way that the devil behaves. To people who are committing evil acts but are justifying them, it would be counter productive for the devil to confront them. In fact the devil will want them to continue to believe that what they were doing was good. Further, to those who were damned, it is beneficial for him for them to remain damned than to confront them with their sin and have them turn around and become saved. As such the whole episode with this character as he devil convincing Markheim that he is actually an evil person despite Markheim believing otherwise is quite uncharacteristic of him. It is more likely that this character is angelic as opposed to diabolical (and remember, the person never actually says who he is, and it is left up to us, along with Markheim, to assume).