Roguelikes are a particular subgenre of Role Playing Games, so named for being like Rogue, a very early computer game. The most traditional roguelikes have the following characteristics:
Roguelikes are centered around Dungeon Crawling through randomly-generated environmentsrandomly stocked from a huge list of monsters and items. Some (such as ADOM) also have a static overworld and/or special levels, but even those games rely on random content in other places. This means that memorization is not enough to win a roguelike, and walkthroughs as such cannot be made for them, but they have high replay value. Furthermore, luck is always a huge factor, as the random elements can turn a good situation into a very bad one (or vice versa) at the whim of the Random Number God.
Roguelikes have only a single controllable character, with a turn-based engine in which everything moves at the same time. Some allow you to have allies or pets, but they can't be directly controlled, only given general orders. The power of your character, companions, and items are crucial, so roguelikes tend to be extremely heavy on Level Grinding and Min-Maxing.
Most roguelikes have randomized appearances for items. In one game, for example, potions of healing might be green potions, and in another purple. Items must be identified either by blindly using them and see what happens (beware of Poison Mushrooms) or by careful observation - or by using scrolls of identify. It's typical, after dying, to be revealed that you had an item which could've saved you, but was unidentified at the time. They might also up the ante by using Randomly Generated Loot.
There are Monster Closets, called "vaults" in the genre, though typically they don't open up on their own; either they have (often hidden) entrances, or you have to dig your way in.
Roguelikes, especially the well-known or popular ones, have often been under continual development for many years, making them extraordinarily large and complex. Many have to use both capital and lowercase letters to have enough inputs for their commands, and some go even further. Interactions are also often very intricate; The Dev Team Thinks of Everything is named for a catchphrase among the NetHack community.
As a result of the above points, roguelikes are mostly very hard. Death is expected to be fairly frequent, enough so that the community has developed the acronym "YASD," for Yet Another Stupid Death. It is easily possible to play many roguelikes for years without even coming close to victory.
Most roguelikes have little more than an Excuse Plot. Some have less. A rare few have more.
Traditionally, most roguelikes have ASCII or similar text-based graphics (the player, for instance, is typically represented by the character '@'), although support for graphical tiles has become increasingly common.
Roguelikes can be roughly classified into a few different Subgenres that occasionally overlap:
Hacklikes, influenced mostly by NetHack. They mostly focus on Dungeon Crawling, and have mostly finite resources to force the player to manage them well.
Bands, influenced by Angband. Bands usually feature a non-permanent dungeon, infinite resources and very tough bosses, so the games are focused on taking levels in badass until the player is ready to punch dragons to death.
Open worlds that typically have content beyond simple Dungeon Crawling, such as multiple quests and a nontrivial plot.
Coffee break roguelikes are roguelikes that are simple and easy to pick up and play for a while (recently renamed "Roguelites" by some fans).
Open-sandbox type hybrids of the genera popularized by Dwarf Fortress, though not all Dwarflikes are necessarily roguelikes.
Though they have a very steep learning curve, many roguelikes are incredible time sinks, which is only exacerbated by the fact that most of them are entirely free.
In the west, roguelikes are mostly a niche thing, with Retro Gamers and gamers who are old enough to remember the likes of NetHack firsthand being the primary audience. There are a few that have managed to creep into the mainstream, however; such as Diablo, Spelunky and Dungeons of Dredmor. Oddly enough, while roguelikes are more readily associated with Western RPGs, the gameplay style is much less niche in Japan, resulting in quite a few Eastern roguelikes as well.
See also Multi-User Dungeon for a related genre of RPG with its roots in Text Adventure games.
Notable games in this genre:
3059, 3069, 3079 and 3089 sit in a grey area between action RPG & roguelike while continuing the theme of random quests, enemies, items (that can be customized with random parts), terrain & more.
Diablo and itssequels, which take the Roguelike formula into real time. It's also more lenient-rather than being permanently killed, you're teleported back to town with no equipment when you die, but with your level and everything in your personal chest intact. it also spawns an entity called "your corpse" on the spot where you died that has all your goodies on it. They have arguably become a Genre-Killer in that almost all new post-Diablo roguelikes take inspiration from it instead of Rogue itself. Its own clones include:
Dungeons of Dredmor not only has sprite graphics, but also animations, sound effects, background music, Difficulty Levels and the option to turn off Permadeath, all of which are very rare for roguelikes.
Monster Gate 1 and 2, two GBA games that function very much like the Mysterious Dungeon games, but only had a Japanese release.
The arcade game that these are based on, where you put in real currency to get game money which is used to pay the dungeon fee for each dungeon (and to cast spells). Each dungeon you start at 0 XP, but can usually take up to 10 spells with you. The game also featured a non-interactive multiplayer where you could beat dungeons to take them over, and the ability to customize your own dungeons (set the number of levels, type of enimies, and specials) and challenge other players to try and beat it.
Mysterious Dungeon (Fushigi no Dungeon) games, all but one of which are licensed spinoffs of other franchises:
RedRogue: A Homage to the Trope Namer involving the now widowed lover of @ guided by his revenant to retrieve the Amulet of Yendor and restore him to life. Unlike the original it is in a side-scrolling platformer format with no jumping. Combat system derives from a rudimentary casting and enchantment system with dual-wielding a main weapon and a throwable weapon.
Risk Of Rain, another hybrid of a roguelike and an action-platformer, this time in space.
The tabletop The SPLINTER, takes the tabletop RPG elements that made Roguelikes Roguelikes and brings them full circle: randomly generated dungeons, a large variety of (very bizzarre) enemies, a focus on (randomly generated) gear for survival, frequent and permanent character death... It feels more like playing a roguelike than playing a tabletop.