What is the name of Sherlock Holmes' arch-nemesis?

The name of Sherlock Holmes's arch-nemesis is Professor Moriarty.

Professor James Moriarty is Arthur Conan Doyle's fictional criminal mastermind and Holmes's most formidable intellectual opponent. Holmes describes him as the “Napoleon of Crime,” a strategist who directs criminal operations from behind the scenes rather than committing every offense personally.

Moriarty first appears directly in “The Final Problem,” published in 1893. In that story, Holmes explains that Moriarty stands at the center of a vast criminal network. Their confrontation at the Reichenbach Falls leads to Holmes's apparent death, making Moriarty central to one of the best-known episodes in the canon.

A common mix-up is assuming Moriarty appears throughout the original stories as Holmes's constant nemesis. In Doyle's canon, he appears directly only twice, although later films, television series, novels, and comics greatly expanded his role.