I was introduced to Neil Gaiman through his work on The Sandman, a British-American comic published by DC Comics under their "Vertigo" imprint from 1989 to 1996. That series is easily one of the best pieces of sequential art that I have ever read and I've been a fan of Gaiman's ever since. Gaiman has this ability to balance the fantastical with the horrific and humorous, sprinkling in allusions to the works of Shakespeare, fairy tales, and mythology. Honestly, I don't believe I've disliked a single thing he's ever done. I've liked some more than others, but I can't remember hating anything he's made yet. Those looking to check out Gaiman's works should probably pick up the first trade of The Sandman, called "Preludes & Nocturnes." Its where I started. Both American Gods (an urban fantasy story about the conflict between the deities of mythology and the new "American" deities) and Stardust (a modern fairy tale about a young boy who promises to retrieve a fallen star to impress the girl he loves) are great places to start as well.
ROBERT E. HOWARD
Robert E. Howard is probably one of the most influential fantasy authors who ever lived. He created what many perceive as the quintessential Sword & Sorcery protagonist, Conan the Barbarian, and inspired similar creations over the years. Whenever I think about great adventure stories, I tend to think about Howard's works and I always want my adventures for D&D and Pathfinder to feel like a good Howard story. While most people seem to only know him for his Sword & Sorcery works, he could also write a mean Horror and Western story. While its not consider one of his best stories, I feel like "A Witch Shall Be Born" is a decent introduction to Howard's works, with "Red Nails" and "Beyond the Black River" being good follow-ups. "The Black Stone" and "Pigeons from Hell" are great if you're looking to check out his horror works, and "Red Shadows" or "Wings in the Night" do a decent job introducing Howard's other character Solomon Kane (the character I actually prefer over Conan).
I know its strange for me to talk about H.P. Lovecraft in a segment called "Fantasy Fiction Tuesday", but here me out. Like Howard, Lovecraft was an incredibly influential writer within the horror genre, creating the Cthulhu Mythos and inspiring numerous writers over the years. However, you'd have to be blind to not see his fingerprints on certain corners of the fantasy genre. The numerous alien creatures that have found their way into fantasy game bestiaries over the years should be a testament to that. While his writing ability is somewhat antiquated by today's standards, the cosmic dread he infused into his stories is still effective to this day. For Lovecraft neophytes, I suggest starting with either "The Outsider", "The Call of Cthulhu", or "Dagon".
I'm somewhat ashamed to admit that I'm a late comer to the Moorcock bandwagon. I read The Weird of the White Wolf last year and I've grown to love Moorcock's writing and his anti-hero Elric of Melniborne. I love how his Elric stories offer interesting advenures while subverting some of the stereotypes associated with the Sword & Sorcery genre. I also enjoy how he finally helped me understand the cosmic interpretation of alignments and how his stories are connected across an entire multiverse. I love how interesting he makes both the big and small details. Since it collects the first Elric stories, I think The Weird of the White Wolf is a great introduction to him. If you can track down the Corum stories as well, those are a great read as well.
While these are probably the ones I feel the most influenced by, they aren't the only authors I enjoy. I'd also list Clark Ashton Smith, Edgar Rice Burroughs, Jim Butcher, Stephen King, and a few others. What about you? What authors do you count among your favorites?