The main resolution of the story is Dupin's realization that the letter is "hiding in plain sight" in D.'s rooms and the trick he plays on D. to take the letter and replace it with a facsimile. Dupin outsmarts D., who outsmarted the Prefect.

In another sense, if you read the story as a kind of argument, the resolution comes with Dupin's explanation about intellect and understanding. According to Dupin, the Prefect's minute search of D.'s apartment was doomed to failure not because of any lack of thoroughness but because of a lack of imagination. Had D. thought to conceal the letter inside the hollowed out leg of a table, the police certainly would have found it. D., however, took into account the Prefect's methods when thinking of a hiding place. He knew that the police would ransack his apartment; he even made it easy for them to do so, frequently spending nights away from home. Because he anticipated the Prefect, he chose to hide the letter in the one place they would not look: in the open.

Dupin foils D. using the same logic. Dupin realizes that D. would know the Prefect's methods; he also knew that D. would as a result choose the opposite method of hiding the letter. So in a way the resolution of story has to do with the use of imagination to reconstruct the mental processes of D. in deciding a hiding place. The story suggests that logic alone is not enough to solve mysteries.

The resolution in a short story or piece of literature occurs when the problem that has been troubling the main character or characters is resolved. In Poe's mystery, the resolution comes when the mystery of what happened to the stolen letter is uncovered: detective Dupin finds it hiding in plain sight.

In this story, a police prefect is asked to find a compromising letter written by the queen. It has been stolen by one of her ministers, who is using the threat of revealing its contents to blackmail and control her. The queen, for obvious reasons, is anxious to have the letter back.

The police search every crevice of the minister's apartment, looking in all of the out-of-the-way or hidden places for the letter. They simply cannot find it. At this point, Dupin gets into the act. When he is told he will be paid 50,000 francs if he recovers the letter, he springs into action.

What he does, however, is not to jump into searching, but to put himself into the shoes of the minister and to try to think how he thinks. Dupin realizes that the minister knows his apartment will be carefully searched and that it is futile to try to secrete the letter in a hiding place. Therefore, Dupin concludes that it must be hiding in plain sight. Searching with this idea in mind, he quickly sees it dangling from a string in the middle of the fireplace mantlepiece.

The point of the story is that we often overlook what is staring us in the face.

On the surface, you could say that the detective, Dupin, locates the letter and turns it over to the police. This resolves the main plot of the story (an early predecessor of the detective genre). However, this is Edgar Allan Poe with whom we are dealing. Do you think that there might be more to the story than the superficial? See if you can think more deeply about it. Perhaps you have more information regarding your assignment than you have expressed in your question and this might help you get started.

I would say that there are two resolutions to Poe's detective story. The first is that (in Poe's opinion) imagination and feeling are far more valuable characteristics to possess than are logic and mathematical skills.  Poe advances this resolution not only through Dupin's lengthy diatribe on poetry, but also through Dupin's (a self-described poet) being able to solve the case that the supposedly logical, well-trained police cannot solve.

Secondly, Poe suggests that humans often tend to overlook the obvious and instead rely on the newest science and technology (or method of thinking) when they could simply use common sense.  Readers can recognize this theory through the police's meticulous search of the apartment which results in nothing versus the perpetrator's hiding the stolen letter with other letters out in the open which Dupin, of course, discovers right away.

The story is an excellent example of the battle of thought that existed in Poe's day between the Romantic authors (Poe) and the Rationalists.

You might want to clarify your use of the phrase "main resolution." Typically, a resolution is the final part of a work in which loose ends are tied up and the plot structure is resolved.  However, "main resolution" appears to be more of an argument term referring to the primary statement in a person's case (argument).