Rogue was an old-school game that laid the foundation for the genre, introducing aspects such as permadeath, randomly generated levels, turn based combat, and extreme difficulty. Thus, the games that followed it's footsteps were "roguelikes"
Today there's an ongoing debate about the use of the word, with some preferring to only use the term for traditional (i.e. turn-based) roguelikes. I'm a huge fan of the genre and prefer the more inclusive use including games such as Spelunky and Towerclimb (roguelike platformers), Tower of Guns and Fancy Skulls (FPS roguelikes), and action roguelikes like Teleglitch and rogue-"lites" like Rogue Legacy.
Roguelikes are randomly generated RPGs where you will play by pushing your luck. The closest analogy I have come up with is that it is like visiting the casino. At which point do you retreat from the dungeon with your treasure? The next NPC or trap could kill you.
Also generally, unlike modern RPGs, Roguelikes generally have no story beyond raid the dungeon to find the Amulet of Yendor. Along the way you find treasure, but you must escape back out to enjoy your treasure.