Call of Cthulhu by H. P. Lovecraft. It's the seminal cosmic horror short story featuring the Cthulhu Mythos. The majority of horror gaming is dominated by the Cthulhu Mythos, so reading the source is a good idea. Many of H. P. Lovecraft's other stories are worth reading as well. Incredibly racist writer which comes through in a few of his stories. Some notable shorts are: Shadow over Innsmouth, Color Out of Space, Music of Erich Zahn, At the Mountains of Madness, Case of Charles Dexter Ward, Pickman's Model, Crawling Chaos, Dunnwich Horror, Herbert West—Reanimator, and so many more.
The Tower of the Elephant by Robert E. Howard. It's one of the most famous Conan the Barbarian stories. The author is responsible for the barbarian archetype we have in RPGs and he's a damned fine writer of action-adventure stories. Like HPL above, Howard was a racist and it mars some of his stories. Many of his other stories are well worth the read. The Frost Giant's Daughter, The God in the Bowl, Shadows in the Moonlight, Red Nails, and Queen of the Black Coast among so, so many more.
Beyond Lies the Wub by Philip K. Dick. One of the best New Wave sci-fi writers. Also read The Gun, Minority Report, and We Can Remember it for You Wholesale. There are many, many short story collections of his stuff. Great all around. His novels are well worth reading, especially Do Androids Dream of Electric Sheep, Man in the High Castle, Three Stigmata of Palmer Eldritch, and Ubik among so many more.
Dangerous Visions anthologies. The third and final entry in the series was recently published after decades of delay and the death of Harlan Ellison, the anthologist and writer who kicked off the series. All well worth the read.
I Have No Mouth and I Must Scream by Harlan Ellison. One of the most controversial writers of recent times and also one of the undisputed masters of short stories. This piece is a pitch black sci-fi dystopia. His other worthwhile short stories are too many to count, but a few are A Boy and His Dog, Repent, Harlequin!" Said the Ticktockman, and Beast that Shouted Love at the Heart of the World.
I, Robot by Isaac Asimov. It's a collection of short stories with a frame. They're all stellar. Also Nightfall is a fantastic story. Just about anything Asimov will be worth the read.
The Veldt by Ray Bradbury. Also read his short story collections The Illustrated Man, R is for Rocket, Dark Carnival, and October Country. Absolute master of the short story.
Scanners Live in Vain by Cordwainer Smith. Any discussion of the story would ruin the impact. Just read it. It's out there sci-fi at its finest. Most of Smith's Instrumentality of Man stories are worth reading.
Virtual Unrealities by Alfred Bester. It's an anthology of his shorts, not a short itself. All well worth the read. Bester also wrote Stars My Destination and Demolished Man, two absolutely brilliant novels.
Green Hills of Earth by Robert A. Heinlein. Another master of the short story. It's the only title I could think of from his short stuff. Pick through his works, you'll find something to love. His novels Starship Troopers and Stranger in a Strange Land are also well worth the read.
I'll legit be here all day, so here's some other authors to check out. All highly recommended. Agatha Christie, Arthur Conan Doyle, Jack Vance, Edgar Allan Poe, Shirley Jackson, Ernest Hemmingway, Robert W. Chambers, P. G. Wodehouse, Franz Kafka, Oscar Wilde, Mark Twain, Charles Dickens, Brothers Grimm, Kurt Vonnegut, H. G. Wells, Dorothy Parker, and so...so many more I can't remember right now.