Poe's collection of short stories Tales of the Grotesque and Arabesque was issued circa December 4, 1839. His publishers Lea & Blanchard limited its production run, despite protest from the author. It was his first collection of prose, though he had previously published a novel and, well, a controversial textbook on seashells in addition to a couple books of poetry.
Poe's early works were somewhat sensational and he was occasionally criticized for "Germanic" stories, what we might call Gothic romance today. In Poe's introduction to the book, he responds to those accusations with the now-famous line, "I maintain that terror is not of Germany, but of the soul." In fact, I believe that's exactly what Poe was doing throughout his career: mirroring the general thoughts and feelings of his reading audience (okay, except for that seashell thing).
Burton's Gentleman's Magazine announced the book's publication, noting that its title "pretty well indicates their [stories'] character." In case you're like me, you might need more help on the terms "Grotesque" and "Arabesque." They are architectural terms, after all, and not often used in literary discussions today. Poe, however, might have been inspired by fellow writer Sir Walter Scott, who used the terms in his essay "On the Supernatural in Fictitious Composition."
Both terms refer to a type of complex Islamic art style used in walls and carpets, especially in mosques. Poe used "arabesque" in this way in his essay on "The Philosophy of Furniture" (what? You haven't read Poe's tips on interior design?). In the case of this book, however, Poe may have been trying to create categories within Gothic fiction much like the categories within architecture.
Some scholars define the "grotesque" stories as those where the character becomes a caricature or satire, as in "The Man That Was Used Up," a story which questions the nature of a war hero (likely modeled on General Winfield Scott). The "arabesque" stories are often psychological as in "The Fall of the House of Usher." Other scholars say "grotesque" means blood and gore, while "arabesque" means terror on a metaphysical level. Ultimately, there is no way to accurately define Poe's intentions for these terms, and scholars who attempt a laundry list of "grotesques" and "arabesques" often end up with different results.
*The picture is an arabesque-style tile from the Wazir Khan Mosque in Lahore, Pakistan, circa late 18th century.