What can one say? Dungeons and Dragons is the tabletop role-playing game that started it all. You play as a heroic character in a magical fantasy setting where monsters of all kinds inhabit an enormous world. 5th edition in particular is a well-made system that is easy to learn and fun to play. If you don't know where to start, start here!
Also known as WoD, NWoD, CofD, White Wolf, and probably more, World of Darkness is a modern era game where secrets lurk just beneath the surface. WoD is the base game, but expands into new realms depending on the type of supernatural foe you wish to run. This system places a heavy emphasis on telling a great supernatural story.
Pathfinder
When you want rich fantasy and min/max dreams
Fantasy
Pathfinder is a truly rich and impressive role-playing game that is a worthy successor to the third edition of Dungeons & Dragons. If you're looking for a modern take on a more classical D&D experience, Pathfinder will not disappoint. While more rules-intensive, players will be rewarded for understanding and playing their character's class to the best of its potential.
The iterations of this system have been many, but there's nothing like holding a lightsaber and knowing how to use it. Correct Disney and George Lucas 90's canon by shooting Greedo first and saving the galaxy from destruction. Star Wars is a great game to jump into because everyone already knows something about this universe. Take that starting point and hyperjump into a galaxy far, far away.
Hackmaster
Humor wrapped inside a death trap
Fantasy
Beginning as a comic strip and a parody of D&D, Hackmaster has emerged as a system in its own right. While definitely not for amateurs, Hackmaster is a brutal but hilarious game that rewards players who use their heads. Swords hurt. Falling boulders kill. You have to work as a team to have any hope of getting out of that dungeon alive. Also, the honor system is flippin' hilarious.
GURPS
Any genre, any era
Science Fiction
Produced by famed designer Steve Jackson, GURPS has one of the most detailed character creation designs in the tabletop world. That detail, however, opens up genres and eras to support any roleplaying game you can dream up. Superhero battles, space rangers, detective film noir, wild west, alien invasions, or even time traveling, the Generic Universal Role Playing System can handle it all.
Longstanding but underrated, L5R presents a roleplaying storytelling system with an Asian theme. Travel back to medieval times in the Far East, swing swords, delve into the mystic arts, and encounter foes from this world and beyond.
The game mechanic is a 2d6 roll for contested or difficult situations. Yet within that simple mechanic, PbtA has spawned dozens of roleplaying games of all types. Ready to simplify the rules and escape the D&D mold? PbtA has your ticket.
Exalted
High octane swords and sorcery
Fantasy
Set in a fantasy world when gods still walked the earth, Exalted lets you play a comic book style superhero capable of amazing feats. Defeat fantastic monsters, conquer kingdoms and serve the gods in an uber-high fantasy world.
Now rolling into its 7th edition, Call of Cthulu is your go-to game when you want to get weird, creepy, and scared. Fans of mystery and people who don't get TOO attached to the lives of their characters will find a warm, slimy place to enjoy a good horror game.
FATE
Simple rules, great games
Fantasy
Science Fiction
Modern
FATE is a flexible roleplaying system that's easy on the game mechanics. Skill roles fall within a set range, known as "bounded accuracy," to keep rules simple and fun. FATE is actually a base game for many other games, each which have some bells and whistles to suit that game's flavor. The custom + and - dice are not required but highly recommended!