Simply, a periscope allows the crew of a submarine to see above water for navigational and combat purposes without actually lifting the submarine from underwater and thereby exposing it to danger.
There is no set range limit for a periscope; it's affected by many factors, weather conditions playing a large part. In addition to weather, the height of the periscope above the water, sea state...
The periscope is used by the submarine to look around the surface of the water, while the rest of the submarine is under water. It makes the submarine harder to find and destroy by enemy...
There are generally 2 periscopes, the Primary and the Attack scopes. The Primary periscope is normally used during normal periscope depth observations of the ocean surface (to verify known Sonar...