Exposure is determined by 3 things: shutter speed, aperture and iso value.
All 3 must come together to give you the correct exposure for a given scene. Example:
Correct exposure for a top lit subject on a sunny day using the Sunny 16 rule is f16, 1/125 using ISO100 film. This exposure value however, is also equivalent to f11 at 1/250, f8 at 1/500, f5.6 at 1/1000. These exposures all represent the same amount of light.
Think of it as a tap filling a bucket. How big you open the tap is the how big your aperture is. For a given bucket (ie exposure for a given scene), if you fully open the tap, the time taken to fill the bucket is short (fast shutterspeed). If you close the tap to allow only a trickle to flow, you will take a long time to fill the bucket (slow shutterspeed). However, the amount of water in the bucket (amount of light hitting the sensor/film) is the same in both cases.
Whether you want a big/small aperture with the corresponding fast/slow shutter speed depends then on the creative choices you make, ie how much Depth of Field you want, how much motion blur you want, etc.
Aperture priority is a mode where you choose the aperture you want and the camera automatically assigns the correct shutterspeed to give you the correct exposure as dictated by the camera's meter.