Digital video cameras optical zoom vs lenses If you read at digital video cameras specifications, you will probably read that camera has 20x optical zoom or even greater. 

If you compare this camera with my Nikor 18-200 lens, the lens itself is much bigger then digital video camera, but digital video camera has bigger optical zoom. How this is made, and why my lenses are so big in compare with digital video cameras?

Term digital video camera is referencing to devices which is used for recording video in first place.
Your facts mislead you. Lens magnification and size have nothing to do with video or stills. They have to do with sensor-size.

Your DSLR has a large sensor and requires a large lens. The digital video camera you refer to has a small sensor and requires a small lens. You can very easily pick some random examples to show the opposite! Take a Canon Powershot SX50 HS, for example, which has a 50X optical zoom and a Sony NEX FS100U which has a sensor larger than your Nikon and will require an even larger lens to achieve the same magnification. This makes it impossible to get the kinds of zoom seen on small cameras without the lens requiring its own transport mechanism!