Nethack is one of the oldest games still being developed, and some consider it to be the greatest game ever made. And no, you don't have to know how to hack. It was based on its predecessor, Hack—named after the hack-and-slash genre of role playing games—and collaborated on over the (Inter)net, and hence: Nethack. It is not, however, a Web-based game.
Nethack is a classic single-player (randomly generated) dungeon exploration game that runs on a wide variety of platforms, and yet each implementation uses the same game engine. But whether graphics or text-based, emphasis is on discovering dungeon detail rather than just killing everything in sight.
Your goal in this massive dungeon crawl is to find the Amulet of Yendor, at the bottom of the dungeon, and sacrifice it to your deity. You will then ascend to demi-god status.
The graphics, or rather lack thereof, are surprising and may turn-off new players. The game's origin was text-based, with @ representing your character and various other symbols representing monsters and items on the ground. Of course, if you can't handle the heat of the sometimes confusing text-based graphics, you can always download the tiled version, with real graphics.
However, if you can look past the primitive graphics the gameplay is exceptionally rich and deep, and the development team seems to have thought of everything. For example, if you are burdened and attempt to go down stairs, the game will likely inform you that you just fell down the stairs. Another (more famous) example involves the many uses of the deadly monster, the Cockatrice. Its touch can turn the player to stone, so attacking it using the player's bare hands is not recommended. If the player does manage to kill one, the corpse may wielded as a weapon to turn other monsters to stone, if, of course, the player wears gloves. Furthermore, if the player-character is female and is poly-morphed into a cockatrice, the player can lay cockatrice eggs, which have several interesting applications. Of the most commonly cited stupid ways to die—and, in my opinion, the most funny—is wielding a cockatrice corpse while burdened, then fall down a staircase and land on one's own cockatrice corpse.
If you haven't played this game yet, give it a try. After the first confusing minutes, this game can occupy many enjoyable hours of play time. Click.
Graphical versions also available for both Mac and PC, though you may need the skills of a hacker to compile them.
Cheers to Nate for first suggesting this one several months ago. =)