What is meant by sun's path through zodiac I read somewhere that sun's daily course and complete cycle is visible in the ecliptic belt by zodiac constellations.  

I didn't understand what that means closest I could find online is: https://www.youtube.com/watch?v=x1DqwC7lRpk  

Is the sun's route visible during the night in some areas? Can someone explain?
The stars are not visible during the day, but they are still there (we just can't see them because of the brightness of the sun). We can't see the stars, but we can calculate where they would be if we could see them.  

Over the course of the year the sun appears to move relative to the stars. This is actually due to the orbit of the Earth around the sun. The apparent motion of the sun is quite simple: it moves around the sky in a great circle, always following the same path, and taking exactly one year to complete. The route it takes is called the ecliptic. And most of the constellations that lie on the ecliptic are called the zodiac (they are very important to astrologers, but not especially important to astronomers).

The sun's route is not visible to the naked eye from Earth, but it can be calculated with great precision.