Since the advent of Dungeons & Dragons, there have been scads of roleplaying games exploring other genres than heroic fantasy. A few have gained serious popularity, like Call of Cthulhu, a horror roleplaying game set in the worlds of writer H.P. Lovecraft, and Vampire: The Masquerade, a roleplaying game that lets players assume the persona of an angsty vampire, powerful, misunderstood, and tormented by dark desires — an obviously potent stew for teen goths.
But I think D&D is by far the most popular not only because it was first, but because it’s the most accessible. The setting is by default a generic medieval one with monsters and magic and feudal government. Fairy-tale stuff: kings, knights, princesses, dragons, gold, gems, swords, and armor. While there are many character types, they’re all basically permutations and admixtures of four main classes: priests, thieves, fighters, and wizards (more generically called clerics, rogues, fighters, and magic-users). Characters can be of various races, including humans, elves, dwarves, halflings (hobbits), and gnomes. (And dragonborn and tiefling and a few others, but I consider them highly optional.)
The trappings of D&D run broad and deep, but really you were already exposed to the essentials before you learned how to read. We understand fantasy archetypes really easily. We’re evolved to understand monsters, magic, and monarchs — we have strong cognitive biases that make us susceptible to the idea of them. That’s really all you need to go on with as far as D&D trappings.
You can get deeper into the essentials of D&D here, in my discussion of what I call “tier 1” D&D.
Next: Attributes and the D20