Short answers, pulled from the story.
Appendix N is a reading list Gary Gygax published in the 1979 Dungeons and Dragons Dungeon Master's Guide, titled "Inspirational and Educational Reading." It names 28 authors, 22 specific books, and 12 different series as sources for the game's concepts, tropes, spells, and monsters. The term now describes a loose literary aesthetic of pulp fantasy and planetary romance from the early-to-mid 20th century.
Gygax singled out six authors as carrying the deepest influence over D&D. They were L. Sprague de Camp and Fletcher Pratt, Robert E. Howard, Fritz Leiber, Jack Vance, H.P. Lovecraft, and A. Merritt. Tolkien also appears on the list but Gygax described his impact as "minimal."
Jack Vance's approach to magic directly shaped D&D's spell system, now called the Vancian magic system after his work. The method by which magic-users memorize spells and lose them upon casting comes from Vance's fiction. Vance also inspired the Thief class and several early D&D spells.
Elan Merchandising served TSR on behalf of the Tolkien Estate, demanding half a million dollars over D&D and a Five Armies game. TSR was told to remove Hobbit, Balrog, and Ent from the game, along with other terms, most of which were later ruled public domain. Separately, the first edition of Deities and Demigods included characters from H.P. Lovecraft's Cthulhu Mythos and Michael Moorcock's Melnibonean mythos, both removed after legal threats.
Despite Gygax describing Tolkien's impact as "minimal" and the Ring Trilogy as "tedious," Tolkien's influence is evident throughout D&D. The Halfling race was originally called the Hobbit, and the Ranger class was modeled after Aragorn. The division of elves and the ubiquitous presence of orcs also trace back to Tolkien.
The Old School Renaissance, or OSR, is a movement of roleplaying game designers who returned to the aesthetics of original D&D. Appendix N was instrumental in this movement; most notably, Joseph Goodman read every book on the list to create Dungeon Crawl Classics. Jeffro Johnson's book on his journey through Appendix N was nominated for the Hugo Award in 2016.