As well as the fully automatic green Auto mode, all DSLR's have several other very useful exposure modes. Aperture Priority and Shutter Priority Modes are two of the most popular semi-automatic modes. But what many people don't realise is that generally speaking, these two modes will give you exactly same exposures. For example, take a look at these settings which show equivalent settings (assume that the current light level is giving a reading of aperture f4, shutter speed 1/250th of a sec):-
Every one of the above settings will give you the same exposure, for example, compare f5.6 at 1/125th of a second with f11 at 1/30th of second. f11 is 2 stops smaller than f5.6 (so less light is let in), and so to compensate for that, the camera chooses 1/30th of a second, which is 2 stops slower than 1/125th (shutter is held open longer).
Also, the only difference between aperture priority and shutter speed priority, and the clue is in the name, is which setting you have direct control over, either the aperture or the shutter speed.
So that leaves a burning question. If it makes no difference to the exposure, why then choose aperture priority over shutter speed priority or vice versa? Well, it all comes down to the creativity element of photography. As well as controlling exposure, the aperture is used to control depth of field, that is, how much of the image is in focus from front to back, and the shutter speed is used to control motion, so whether you want to freeze the action or show movement.
With that in mind, for landscapes you would generally use aperture priority and choose a small aperture to give you a large depth of field which in turn will ensure everything is in focus.
For portraits, aperture priority again then choose a largish aperture, for shallow depth of field to blur the background.
In low light situations when you have the flash turned off, it’s usually best to use aperture priority and set a large aperture to ensure the maximum amount of light through the lens. But watch your shutter speed doesn't drop to a slow value, otherwise you'll end up with a blurry photo. If it does, raise the ISO to compensate.