Recently, a friend gave me a gift copy of a Humble Bundle, described as "Roguelike games". Roguelike is a surprisingly widely-used adjective for games, and to listen to many people, often misused. I'll admit, I'm one of those who thinks it's often used incorrectly, but I really need to justify that. Let's get on with it.
I can speak with a certain amount of authority on what Rogue itself is like; it was, twenty-mumble years ago, the first computer game I ever played. I never did win it, but I played it a lot. It had a lot of things going into its makeup. I've also played a lot of games considered lineal descendants of Rogue: NetHack, with forays into SLASH'EM; ADOM; several variants of Angband; and many other roguelikes, most of which I can't even remember the titles of. Everything from a futuristic one set after the end to an educational one meant to teach the country top-level domains. Frankly, for as much as I love other games, roguelikes are the defining games of my life.
But which elements of Rogue make it distinctive? Here's where opinions diverge; the best I can do is give my opinion, and ask that you use your own judgement.
First: turn-based gameplay. When playing Rogue, time does not pass in the game until you provide input. This lets you think tactically, and also lets you gain a deep appreciation of just how hosed you are when you get surrounded. To me, this gameplay element is essential for a Roguelike.
Second: quantised movement. Generally, this is a square grid, but I'd consider hexmaps acceptable as well if anything used them. This originally came about as a consequence of Rogue's being used on a text-only display, giving it a grid to work in, but it's also a natural consequence of the turn-based gameplay. Another oddity is that a rat and a dragon take up the same amount of space. Typically, any given level is generated by a random number generator, but there are often pre-defined levels.
Third: inventory management and unpredictable inventory contents. The unpredictability is in the effects of a given item; what a yellow potion does this time around has nothing to do with what it did last time. You'll have to suck it and see, unless you can find a means of identifying things. Meanwhile, you'll be carrying around a huge amount of items.
Fourth: saved games are for taking a break. You can save your game; this ends your game session. When that character dies, his save file is done. You start again at the beginning.
Fifth: not taking itself too seriously. Roguelikes are written by geeks, for geeks, and geeks love inside jokes and oblique references. Monty Python, Dungeons and Dragons, Tolkien, and just about anything else are targets for a gentle ribbing, and so is the game itself. Rogue was fairly serious, but the scale goes as far as Dungeons of Dredmor, which takes almost nothing seriously; sit idle for a while, and your character will pull out a phone and start texting.
By these standards, nowhere near all of the games I was given qualify as roguelikes by my standards. One that does, though, is Sword of the Stars: The Pit, which eschews the traditional ASCII graphics in favour of nicely drawn sprites, but is otherwise a very impressive science-fiction Roguelike.