For those of you who can't wait for the elaborate answer, here is the quick overview:
Commercial Photography sells a product.
Fashion Photography sells a lifestyle.
Editorial Photography sells a story.
Now for those of you interested to learn more, please read on. Just take note that the borders between those different types of photography are fluent and may not be as distinct as I describe them here. Nonetheless, I am sure that this article will help you to easily distinguish between them. First of all let me tell you that Fashion Photography and Commercial are usually summarized in the category of Advertising Photography, while Editorial Photography tends to be a category of its own. The reason being is that both Commercial and Fashion foremost try to sell a certain product, while Editorial Photography is more concerned with a story.
Commercial Photography As mentioned earlier, commercial photography predominantly sells a product. Hence, the emphasis of the whole shoot is on the product. That means that the lighting, the styling, and the background are usually very plain (not in a bad way). Think of catalog shoots where the models is standing in front of a white or an off-white background with maybe one light above camera and slightly to either side of the camera, and very natural looking make-up.
Fashion Photography Contrary to commercial photography, the emphasis in fashion photography does not lie on the garments, but on the mood and styling of the image. The garments are merely an accessory to convey a certain lifestyle. Therefore, the whole image is much more complex. You usually won't see plain white or off-white backgrounds in fashion photography and the models are usually styled very dramatically with thick eyeliners, dramatic eye shadows, etc. With the increased complexity in styling also comes an increased complexity in lighting to make the image look more dramatic. While many commercial photographers only stick with one light for the model, fashion photographers tend to use a wide array of lights and lighting accessories. When doing a fashion shoot, I have used up to seven lights to get the dramatic look that I wanted, but I know photographers who use ten or more lights for fashion. That is not to say that a dramatic look can not be achieved with only one light, such as a beauty dish or an octabank - it's about the effect of the lighting and not about how many lights you use to get that effect.
Editorial Photography Editorial photography is very similar to fashion photography in the sense that editorial photography is usually not about selling a product but something greater. In fashion photography it's the lifestyle, in editorial photography it's the story or the theme. As with fashion photography, the proper lighting is of utter importance. Let's say you are shooting an editorial about a girl traveling. If the story is about the joy and the happiness, you would of course use very friendly and open lighting to emphasize this mood. If on the other hand the story is about the dangers of traveling, your lighting would be much more dramatic and instead of a beach on a sunny day, you may choose a dark alley.
Conclusion In essence, your style of photography depends on the clients' needs and wants. If the client wants to sell a lifestyle associated with his product rather than the product itself, you will have to deliver fashion photography. If the client on the other hand just wants to show off his products (such as for a catalog shoot), you will have to deliver commercial photography. If the client already has a story or a script and just wants the photos to support the story, it's your turn to get the mood of the story and deliver an editorial piece on it.
A good photographer will be able to deliver quality work in all of these areas. Just don't think that if you are good in one area, you will also be good in the others. Each style of photography requires a different set of skills and a different approach.