Death occurs when your dwarf's hit points is equal to exactly zero. This results in a game over (though if you sired any progeny, they can inherit some of your gear and progress).
Having less than zero hit points does not result in death; only when your dwarf's hit points are exactly zero does death occur. The game's damage is coded never to exceed your maximum number of hit points, and so you will never, as a result of damage, fall below zero. It is possible, however, to start the game with negative hit points (the so-called zombie status), in which case taking damage results in moving further away from death (and healing results in moving closer to it).