Indeed as others have said the 'dangers' of Dwarf Fortress come in a few levels (and as they said, death elephants and carp and the like are things of legend now, no longer reality).
First will be basic survival, did you run out of food? Water (how often has that happened to me.. See all your dwarves all of a sudden have those blue arrows..), did you start a flood in your fort? These are generally what brings down new players. Didn't correctly and quickly set up your food and water supply, another frequent one is relying on rivers and/or lakes only to have them freeze or dry up during parts of the year (though the caverns have made that less of an issue).
The second issues that come up are monsters, after you can survive and get the basics down you'll have to start dealing with monsters. First the occasional random enemy (trolls, blind cave ogres, etc) which can cause problems if you don't have anybody to fight back adequately, then ambushes and finally sieges and forgotten beasts and titans and the like..
The last, as was mentioned, are the 'game' deaths.. FPS is a common one, when the game just gets too slow to really enjoy playing anymore and while there are some tricks to last longer short of a supercomputer there is nothing that will make it go on forever, if you go long enough sooner or later time will grind to a halt. The other is, as comes up often, bordom. Once you get to the point that you can generally set up a fort anywhere without issue, you can tackle all but the biggest sieges and monsters then the game loses some of its luster.. Once you get to that point you need to decide what you want to do. Some options are challenges (such as the 'cavern challenge' lower on the forum, or any self-imposed rules you use to make achieving the goals you can already do easily much harder to impossible), or mega projects if that is your thing. There are also mods that can breath some interesting life into it (for example I use fortress defense II, radically increases the siege amounts to the point its not uncommon to have three or more sieges come at the same time. Even if you do get tired of it I've found taking a few months off brings back the fun, I did the cavern challenge today and yesterday and had a blast but before that its been months since I've played.
So in short the dangers in order of how soon you need to overcome them are basic survival (food, drink, shelter), monsters (creatures, ambushes, sieges, megamonsters), and yourself (bordom, FPS).
Good luck, its an amazing game!