Shutterspeed determines how long the film or sensor is exposed to light. This is achieved by a mechanical shutter between the lens and the sensor which opens and closes for a period of time.A shutter speed of 1/60s will expose the sensor for 1/60th of a second.
Shutterspeeds are expressed in fractions of seconds, typically as (approximate) multiples of 1/2, so that each higher shutterspeed halves the exposure by halving the exposure time: 1/2s, 1/4s, 1/8s, 1/15s, 1/30s, 1/60s, 1/125s, 1/250s, 1/500s, etc. Long exposure shutterspeeds are expressed in seconds, e.g. 8s, 4s, 2s, 1s.
The optimal shutterspeed depends on the situation. A general rule of thumb is to shoot with a shutterspeed above the inverse of the focal length (1/focal length eg 1/200 for a 200mm lens) to avoid blurring due to camera shake. Below that speed a tripod or image stabilization is needed. If you want to "freeze" action, e.g. in action photography, you will typically need shutterspeeds of 1/250s or more. But not all action shots need high shutterspeeds. For instance, keeping a bird in flight in the center of the viewfinder by panning your camera at the same speed as the bird allows for lower shutterspeeds and has the benefit of creating a background with a motion blur suggesting movement and speed.