This synopsis will contain spoilers!
Dr. Watson, having returned from being injured in Afghanistan, needs to find someone to live with in order to afford a place. Discovering this, his friend introduces him to Sherlock Holmes, who has an apartment in mind, but also needs someone to split the cost with. The two readily agree to begin living together.
Watson is confused by the subject matter that Holmes is interested in, and the areas of his intellectual expertise. When Watson scoffs at an article espousing the importance of deductive reasoning, Holmes reveals that he was the author, and finally reveals in detail that his interest is in solving crimes. Watson is impressed by his abilities, but does not quite understand the depth of Holmes belief and skills.
Soon a message arrives stating that Holmes is needed on a case. Watson accompanies him, and they arrive at the scene of the murder where a body, covered in blood but without a wound, lays in an abandoned home. Holmes gets the details of the case, does his investigation, and claims he already knows enough to solve it.
After following up on a few items, Holmes and Watson are in their apartment when the two detectives arrive, each effectively revealing that their theories on the case were incorrect as well as a second murder, this time the secretary of the first murdered man. At that moment, a cab driver arrives for Holmes, and he puts handcuffs on him, revealing that the cab driver, Jefferson Hope, is the murderer.
At this point the novel transitions to telling the story of John and Lucy Ferrier, two Americans who were rescued in the desert by Brigham Young and the Mormons. Years later, Lucy and Jefferson Hope (a gentile) are engaged. When Brigham Young and the Mormon Elders discover this, however, they tell John Ferrier that he has one month to decide which of the Elder’s sons Lucy will marry instead. Jefferson arrives 2 days before the deadline and the three escape into the mountains.
After many days journey, Jefferson leaves to go hunting, and while he is gone John is murdered and Lucy is returned to Salt Lake City where she marries Drebber (the first murder victim). One month later she dies. Jefferson swears vengeance against Drebber and Strangerson (who killed John and wanted to marry Lucy as well). He follows them for years, looking for an opportunity to kill them, finally getting the chance in London.
After revealing all of this to Holmes and Watson, he is sentenced to appear before the Magistrate. However, before that can happen, he dies of a heart aneurysm. Holmes reveals to Watson how he was able to unravel the mystery and catch the murderer. The two detectives, of course, take all the credit for the arrest.
Sir Arthur Conan Doyle the Last Line of in A Study in Scarlet Scott notes: Google reveals a translation of "The public hisses at me, but I applaud myself in my own house, and simultaneously contemplate the money in my chest."