Shutter speed is the amount of time the shutter remains open to allow light through it. An extremely fast shutter speed is 1/2000 of a second, while camera settings usually allow up to about one second, which is very slow. One-sixtieth of a second is about as slow a shutter speed as you can use when taking a hand-held shot, and not get any blur. Some photographers force their camera shutters to stay open for much longer to create various special effects. Leaving a camera pointed at the night sky with the shutter open for several hours results in a photo of the paths the stars seem to take across the sky as the Earth rotates.
Practice and experience are the best ways to figure out which combinations of aperture and shutter speed are best for different kinds of photos. While a slow shutter speed lets in more light, it also makes it very difficult to get a crisp picture. Any movement at all (of either the subject or the camera) will result in blurring. Sometimes you might want this effect, but for a clear photo of a moving object, you need a fast shutter speed.
Many cameras have a semi-automatic mode that can be set to either aperture priority or shutter priority. You can either set the aperture or shutter speed (depending on which priority mode is enabled) to the desired setting, and the camera calculates the right settings to accommodate the lighting conditions. The camera might also have a variety of modes to choose from, such as sports mode or outdoor mode. These are aperture/shutter speed presets. Again, experience will let you know what conditions are right for each mode.