This feature of N is generally used ONLY when there is no way out of your predicament, a situation known as undeath. You can activate the suicide feature by pressing the "K" key in the game, unless you changed the key using the configuration menu. This will cause the ninja to die--either by spontaneously exploding or apparently being sniped and crumpling to the ground/being thrown into the air--and you can then restart the level.
Of course, users that are not simply aiming to complete a given level and are going for highscores may also use the suicide function whenever he/she makes a mistake in a run: oftentimes when highscores are involved, a single mistake marks the doom of any chances of obtaining a score. Thus, to avoid wasting time by completing the imperfect run and not obtaining the desired score, most players would opt to instead suicide and start the level anew, saving time.
Note: Suicide death counts are under the ninja icon in the Kill count section of your Highscores page.