Of course, the best way for you to gain an understanding of what happens in this classic of Gothic short fiction is to read it yourself. However, I hope that the outline I sketch for you know will encourage you to return to the story and dive into it!
The story centres around a character called Markheim, who has gone, on Christmas day, to a antique dealer's shop, supposedly to buy a present for a woman he is trying to court and marry. The dealer and Markheim are in the shop alone, and the setting is described with typical Gothic attention to atmosphere, detail and sound, with the ticking of the clocks and the darkness of the shop evoking an oppressive and foreboding atmosphere. When the dealer offers a hand-mirror to Markheim, Markheim's reaction is one of horror at the suggestion, as he calls the hand-mirror a "damned reminder of years, and sin and follies--this hand-conscience." Clearly this foreshadows the murder that Markheim is about to commit. As the dealer turns to look for another possible gift, Markheim stabs him and he dies.
Markheim pays attention to the way that his surroundings have changed, and in particular to how the flickering candlelight gives an appearance of movement to the static objects. He begins to suffer from tremendous fear and terror, imagining others will have heard the act and contemplating his death at the gallows. Suddenly, knocking comes from outside, but after a while leaves.
Markheim determines to find the money and leave quickly. He regards the corpse of the dealer with pity, but with no trace of guilt as he finds the keys. As he goes up to the office, he becomes obsessed with the idea that he is not alone. He shuts himself in the office and experiences a few moments of peace as he begins his search. However this respite is but temporary as he hears the sound of a person mounting the stairs to the office which fills him with terror. The person enters and Markheim is struck by his strange appearance. The person engages in a conversation with Markheim and tells him that he knows Markheim better than anyone, and goes on to try and tempt him to engage in yet more acts of evil, killing the maid who is about to return. Markheim, after being confronted with the irrevocably evil nature of his life, determines to challenge this destiny by laying down his life and turning himself in. As he does so, the expression on the face of this figure softens "with tender triumph." However this is unseen by Markheim as he advances downstairs and tells the maid to call the police because he has killed her master "with something like a smile" on his face.