When you use the AUTO setting in your camera, the image is analyzed and the internal "computer" will decide what settings it will change. In very general terms, if the camera sees a predominant bright blueish area in the frame, this means you want to shoot a landscape. If it sees an big oval shape, with round objects on the upper half, this is a portrait. With faster and more modern cameras, this technology can even see more than one face at the same time and recognize them if you have a name for them.
Sometimes the camera computer seems a little too intrusive and you may want to override it. There are occasions when you may want to have a different perspective, or you want to be a little more creative and the camera will not let you.
Aperture Priority mode is one of the options that can give you more freedom to create; in Canon cameras this is indicated with the Av icon, in Nikon it is just A . This option controls how open or close the lens will be during the exposure. As a direct consequence, opening the lens will let more light enter into the camera sensor, giving you opportunity to improve your composition in low light conditions.
Another advantage of shooting in Aperture Priority mode is the ability to control how deep your focus is. Wide apertures will allow you to blur the scene background, making it less cluttered and giving it a "smooth" texture, all this to help you put more emphasis on your main subject (foreground). This technique is commonly used in many photography genres like portraits, close-ups or macro photography.
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On the other hand, closing the aperture will give you more of the scene in focus, and certainly less light to the sensor. Landscapes or architectural photography take advantage of very small apertures as they usually present a sharp foreground, midground and background in the same photo.
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One caveat to consider is how the aperture scale is presented; at first sight it is not very intuitive. Just keep in mind that the smaller the number is, the wider the aperture you'll get. There is a reasonable physics explanation for that, but we'll leave it for another time. The scale may go from "f / 3.5" as the widest aperture to "f / 22" as the smallest setting. More expensive lenses can open up to "f / 2.8" or even "f / 1.4"; yes, "f / 2.8" is wider than "f / 3.5"... and "f / 1.4" is even wider than "f / 2.8". Enough with the numbers. A more detailed and boring explanation can be found at http://en.wikipedia.org/wiki/F-number.
Now that you know how Aperture Priority works, don't be afraid to use this option as part of your creative photography. The more you practice, the more you'll be able to get stunning images.