Its basically equal parts Sim City, Nethack, and Starcraft. You have to manage / micro-manage resources (chop trees, mine ore, &c.) to construct buildings, food, weapons, &c. for your colony of dwarves, like Starcraft.
At the same time, each dwarf is a completely unique person--you can access a menu that will tell you how they feel about their co-workers, whether they've admired a particularly nice chair recently, and whom they're in love with. They might move in with someone and have kids, which increases the overall positive vibe of your community. On the other hand, if they get stressed out they might throw temper tantrums, feel sad, wander off, or even attack the others. An overall happy (and prosperous) community attracts more dwarves--like Sim City, your success is rewarded / punished with more people, which makes your job harder (but gives you more resources).
Finally, it borrows a lot from Nethack. Not just the interface (it's naught but ASCII graphics here, with little @s running around getting killed by Ns and |s) but also the basic idea of exploration--it's a vast, vast world, and a lot of the game is simply finding out what's there. It's also an RPG, like Nethack--each dwarf gains in experience and becomes a better miner, a better fight, &c. with practice.
The real Nethack legacy, however, is the ridiculous complexity of the game. The graphics are offputting, but serviceable. The interface, on the other hand, is really awful (endless keyboard commands, non-standard menu shortcut hell, &c. Want to build a kitchen? Just type b, then w, then (I think) y. It's obvious!) but the tradeoff for these fairly minor (but common) niceties is everything else. Just read through the FAQ for examples. Here's one illustrative bit of complexity:
Dwarves love to drink, and if they don't get enough booze they become unhappy. So you should make them a still, and assign someone the job of Brewer. But what to brew? Plants, of course, so you need a farm, and a farmer. But the increase productivity you should really fertilize the farm, which means having yet another dwarf make potash from lye and the ash from the trees that still another dwarf has been assigned to chop down! A far cry from Doom II, let me tell you.
Then there's combat--not only for the goblins that might attack (or the dragons and other megafauna), but also for hunters to get meat (which can then be butchered, rendered into soap, the skins tanned and turned into bags and armor) for supper. If the dwarf is using a crossbow and runs out of ammo, they'll start using their Hammer skill to bludgeon the enemy, instead. Unless they're trained Wrestlers, in which case they'll drop the crossbow and try various immobilizing holds. Assuming that their arms are strong enough--the game models each limb separately in combat, and a nasty cut to the left arm can slow down the hardiest wrestler...
These are not at all the "main points" of complexity, they're just examples chosen at random. Every aspect of the game is just as deep, down the the world itself (which accurately models the various types of rock--one FAQ entry suggested that a High School level geology textbook would be helpful to really play the game well).
Also, the world is completely random, generated the first time you run the game. And it's huge--on older computers this generation took up to half an hour. How big is it? Well, at a very rough guess, I'd say that the level is about five computer screens by five, or about 20 million pixels. (assuming 1024 x 768 resolution). But each level is just a sub-section of a sub-section of a sub-section, one possible choice among a million randomly generated possibilities.
The coolest thing (well, a cool thing) is that the world is persistent. So, just like in Nethack there are no saved games--you can save your progress, which ends the game (so that you are allowed to leave to work!) but when you start again your only option is to pick up where you left off. However, after you die (or give up in frustration, or get bored) you start over again in the same world. This world that now includes your (abandoned, empty) fortresses from previous games, and might have songs and legends about deeds that you've done in other games. In fact, you can play Dwarf Fortress in "Adventure" mode (which I haven't tried) which is just like Nethack, and in the process you might find yourself exploring fortresses from years past.
OH! And I forgot to mention! Each of these frickin' huge levels has a vertical axis, as well! Being dwarves, a lot of your time is taken up mining deep into the earth. Like, ten levels below seawater. You can also go up ten levels above sea water, mostly (I think) by climbing mountains. And things on lower levels interact fully with things on higher levels! I made myself a well--an enormously complicated process that involved drilling the shaft, forging a chain, crafting a bucket, and draining a nearby underground lake to fill it. Only then did I realize that it was merely a lower level of the same above ground lake that one of my dwarves was fishing. Was, that is, until it drained down into a muddy pool.