As explained on the front page, "Roguelike games" are a kind of rather difficult, heavily retro style, turn-based graphical RPG, but normally with text-based graphics (many purists such as myself don't strictly count games without at least a text-based mode as "roguelike", although they can be seen as closely related). Those that have a graphical mode, use tile-based graphics, as everything still fits firmly into a grid.
Due to the random dungeon, you never get the same game twice. But you would (assuming you didn't give up) wind up starting a lot of new games anyway, because whenever you get killed by a monster, that's it, game over, no "continue?", no restarting from your last "saved game" because "saving" is just to let you leave the game for a while, and exits the game, whilst "loading" the game deletes the save. No second chances! This isn't so much why Roguelike games are hard (though they typically are), as why many other RPG type games are easy.
Different games wound up picking different symbols for things, although there were some consistencies- for instance almost every roguelike follows the original pattern that letters are for monsters, although many also add a few exceptions (some games also use "T" for trees, many games also use "&" for demons, etc). Also walls are almost always a # symbol, and empty sections of floor (in a room or corridor etc) are a . symbol (full stop, or "period" for the Narnians in the audience). Ok there's some games that consider rooms and corridors to have different floors. Also potions are typically a ! and scrolls a ?. Some games don't have such things, but use the same symbols for their nearest equivalents!
Many games also add things like skills in various guises, in addition to the stats that Rogue had (actual stats used vary a bit of course), many also add classes and or races for your character to play. All this gives a whole load of variation in what sort of game you can play, even if "wander about and kill stuff" remains a major part. Some in fact, you can technically finish the game without doing, and offer this as a potential challenge to take up!! The fact that many games award experience points for other actions too, help with this but it's still not exactly easy. The "normal" type of play is hard enough already!
Other frequently seen features that (AFAIR) didn't crop up in the original Rogue, include the option of digging through walls and rock. This tends to be slow and/or makes you get hungry much faster, but can be advantageous as you can probably imagine! Also features such as item creation of various sorts, are seen in a few.
One of the most famous families of games are either directly descended from Rogue, or from games very strongly inspired by it. One such game early on was "Hack", which led to the very successful "Nethack". Note that the name has nothing to do with the usual meaning of "hacking", and "Nethack" is not multiplayer or networked (except insofaras you can play the Unix version over Telnet or Secure Shell, like any other text-based game). It's "Hack" as in hacking at monsters or hacking your way through the dungeon, and "Net" because Nethack was developed on Usenet (part of the internet, if that actually needs explaining...) as a collaboration by many amateur developers. Nethack spawned even more offshoots as it was open source with a pretty liberal licence. I think there were more in the past than today. Some sort of languished in obscurity, but "Slash'em" in particular grew to be pretty successful in its own right, and has very cool features.
Nethack type games, like Rogue did, generally have each level of the dungeon as a single screen, but unlike Rogue the layout of each is more random, and you can also return to older dungeons and find them as you left them! Like Rogue, the risk of starvation can be significant, but this is greatly ameliorated by the fact that almost anything you kill, you can eat afterwards. This is not always a good idea though!! But as new monsters continue to appear after you've entered a level, in principle you can survive a good while, if you're lucky.
Nethack also has a lot of interactive features in the dungeon (eg altars to pray at, thrones to sit on, fountains to drink from), and many of these also have obscure other features to make use of (or avoid!!). Similarly, a lot of the objects you carry can have extra uses besides those obvious for the class of object they're in. It's reminiscent of some puzzly RPGs in that respect, a lot of neat little secrets to find out. Some of those features, and some of the monsters, are considered by some too random and silly, but others consider them fun- it's a matter of taste really. Another feature largely associated with Nethack-family games is all(?) characters start with a pet who will follow them around and fight on their behalf. This can really save your bacon as early on it's pretty much tougher than you. As it can gain experience as well, it can grow even stronger. Unfortunately it can also eat food you wanted to eat, particularly if you're not careful, but it isn't usually that bad a problem. You can also tame further pets too, although many others are a bit crap. Still, it gives some extra flavour to the game.
The other major family of games descends from Moria. Or Larn. Or... one of those. They're all related, share a very distinctive look, and ended up yielding the much more successful "Angband" and its descendants. Unfortunately although these (at least as far back as Angband) are open source, the licence forbids people getting any sort of money for the game, which restricts availability some (particularly in many Linux distributions, which would be the natural home for most roguelike games). Yet still, there are many many derivatives of Angband, some of which such as Zangband are I think rather more popular than Angband itself. Whilst the original games are specifically Tolkien-based, some of the variants have another setting/theme (but may keep over some of the monsters anyway). There is also at least one multiplayer Angband variant!
Moria/Larn/Angband type games are noticably different to Rogue in that they start off with the character in a small "town" level just above the dungeon, with several shops. In fact this is very similar to what little I've seen of Diablo (don't be mistaken though- the Roguelike games didn't copy Diablo, having well predated it; it might well have copied them though, as they were pretty well known in the right circles). After getting any further equipment you might need, you can descend into the dungeon, each level of which is big. Around 5x5 screens worth. And whilst you can go back to previously visited levels, the levels are never the same when you do so, you are essentially going to entirely new parts of the dungeon. A good reason never to leave anything behind that you want!! You can still return to the town, but (AFAIR) can't expect to leave things lying around there either. In most games you do have a "house" to store things safely in though.
You don't tend to see much in the way of "interesting" stuff in Angband dungeons besides mineral seams and the occasional "vault" chock-full of very strong monsters and treasure, and there isn't the same versatility and depth to the items. That's not to say the games are "dull", the focus is just different I figure. Like in Nethack you can also tame pets, but nobody starts with one, and AFAIR they can't advance (I haven't used them that much, or recently). Most Angband type games tend to quite strongly emphasize the differences between classes, such that for instance "Warrior" classes can learn no spells, and the spellcasters are firmly limited to specific sets that they can use. Some such as "Sangband" do blur these distintions some.
Meanwhile others pretty much stand on their own. Notable examples would be ones such as ADOM, Crawl (aka Linley's Dungeon Crawl and also the more recent Crawl Stone Soup versions which are sort of a continuation), and Omega for those who still remember it. These aren't derived from others and (unless you count the later versions of Crawl) don't have descendants either. Some like ADOM and Omega are substantially more RPish/story based than many others, with a fair bit of plot happening as you play. There's plenty more such standalone roguelikes I haven't even played, and so couldn't tell you about.
Nethack family:
Angband family, aka *band:
Others: