Many digital cameras let you control aperture and shutter speed. Together, aperture and shutter-speed determine how a photo is exposed.
The aperture, often referred to as the iris of a camera, is the opening in a lens which admits light onto the camera sensor. The aperture can be varied by opening the size of the hole. It's diameter is stated as an f-stop.
Shutter speed relates to how long the camera's shutter stays open after you depress the shutter-release button. Shutter speed is measured in seconds, such as 1/125th. A slow shutter speed produces a longer exposure time.
Aperture and shutter-priority are called semi-automatic modes because you when you manually set the aperture, the camera automaticallly sets the shutter speed. When you manually set the shutter speed, the camera adjusts the aperture.
The reason you would want to use a semi-automatic mode depends on the type of photo you are taking. If you want to freeze the action of a fast moving subject, you'd use a fast shutter speed. If you want what is known as shallow depth-of-field when taking a close-up shot, you'd use a wide apeture.
I know I've used a lot of fancy terms here so I want to give you some references with illustrations. You'll gain a better understanding of just how these settings can influence the the look and exposure of a photo.
Aperturehttp://www.digicamhelp.com/advanced-digital-camera-settings/aperture.htmShutter speedhttp://www.digicamhelp.com/advanced-digital-camera-settings/shutter-speed.htmDepth-of-fieldhttp://www.digicamhelp.com/taking-pictures/depth-of-field.htm