X-rays are similar to light, but they have more energy than light, so they can penetrate more materials than light can. For example, light does not pass through your body, but an X-ray can.

So when someone takes a picture of you, light bounces off of you and bounces back through the lens of the camera. This reflected light forms an image of you on the film. However, when an X-ray image is taken of you, the X-ray machine sends an X-ray through your body and that image forms on the X-ray film.

When the X-rays hit the film, they expose the image just as light would. Since bone, fat, muscle, tumors and other masses all absorb X-rays at different levels, the image on the film lets you see different (distinct) structures inside the body because of the different levels of exposure on the film.

By the way...
Ever wonder how those toys you get in your cereal box glow an eerie green when you put them in the dark?

Glow-in-the-dark items contain phosphorescent pigments that emit light energy and X-rays!

What happens:
First, shine a flashlight on your glow-in-the-dark toy. Right now, you are exciting some of the atoms in the toy.

Next, look at your toy in a dark room. It is glowing because the excited atoms are beginning to calm down. When excited atoms calm down, they release energy. What kind of energy? Whatever excited them in the first place (in this case light, but it can also be other things such as X-rays).

Hey, this sounds like radiation! Is this harmful?

No. They emit energy at such low levels that they're barely detectable. The sun's radiation is much worse, actually.

(Source: The Straight Dope and HowStuffWorks)