As a photographer, you're usually aiming to ensure that your photos are well-exposed and have an even tonal distribution. If you were to look at the histograms from these images, the data would be spread across them. Neither the shadows nor the highlights would be 'clipped', or have areas that are so dark or so light, respectively, that they contain no data. If you were to look at these images, they would look 'right', and be pleasing to the eye.
However, that isn't always going to be the kind of story that you want to tell. On occasion, you are going to want to convey a dark and brooding narrative, maybe even so far as to be menacing, that delves into darkness. You will look to shoot a low-key image.
Identifying low-key images
Low-key photography is a style of taking pictures that focuses on producing shots that mostly include dark tones—a person in a dark room with only a tiny bit of light on their face is a good example. You might have noticed how products are often shot using low-key photography, too.
If you were to look at the histogram of a low-key image, you'd see that the majority of the data are pushed towards the left-hand side. Don't worry that your shadows might be clipped: it's a part of the look.
Setting up a low-key shot
You're aiming for extreme contrast in a low-key shot, with most of the image dark or even fully black with just a little of subject highlighted to draw the viewer's eye. Since a lack of lack is your primary tool in taking a low-key photo, it's actually fairly easy to set up and while you do need extensive control over your light source, don't think that you require an elaborate studio.
One method of getting the low-key feel is to place the subject in a dark room, then turn the light on in an adjacent room. Open the door just enough to get the right amount of light on your subject, but not enough to light up the background.
Another means to getting a low-key feel is with candle-light, which can elicit a suitably gothic feel, or by using a torch. Torches are easy to direct and definitely cheap! If you want to start manipulating light in your photography but don't yet have the means or desire to go the whole hog with flashes or a studio set-up, low-key is a good option.
You will certainly need to be in manual mode in order to achieve a low-key shot. If you leave it to your camera to decide on optimal metering, it will try to over-expose the scene and ruin the effect. Using spot-metering to expose for the sliver of light, wherever it might be, is a good idea, too. This will help to keep that area light but the shadows dark.
The general rule is to keep things dark. Complete blackness around your subject (or even as part of your subject) is perfectly acceptable.
Processing low-key images
Ideally, the combination of manual control and carefully manipulated light means that you won't need to adjust the exposure beyond the odd tweak to the blacks or the highlights to ensure that they're sufficiently dark or light. It is worth considering converting your low-key images to black and white: initially, this can help you to identify and perfect the contrast in low-key images but over all it can help to increase the drama and mood in the photo.
More unusual ways of looking at things, remembering rules, and then breaking those rules, are in my lovely book, The Rules of Photography and When to Break Them. It's available as an e-book and in a dead tree version (UK, US).