Ok,

Going by your Questions I paste below and answer accordingly .

•What is the use of the score?

The use is to determine whether you have an advantage or you are better playing or winning  . For best results take a good game and analyse with Fritz  / StockFish . You would see that as the game progresses there will be a shift as whoever plays well  .

•My understanding is that it depends on the opponent's move against our move. Am I right?

Almost true but not absolutely. The Score is a mathematical evaluation like for example during the start of the Game it is 0.0 . It means both are equal at this point  . The Score depends upon a number of factors like more space, pieces  , king safety and a lot more  .

•What is the minimum and maximum range of the score?

The Score is always in favour of Black or White  . You can see that when it has a winning combination it can be -infinity to +infinity  . The higher the score it means it favours a particular side  .

•When one can achieve the maximum score at very first move?

It does not depend on you except you are playing your opponent at the beginner level and scholar's mate or fool's mate is played on board  .

Ok,

Going by your Questions I paste below and answer accordingly .

What is the use of the score?

The use is to determine whether you have an advantage or you are better playing or winning. For best results take a good game and analyse with Fritz/StockFish. You would see that as the game progresses there will be a shift as whoever plays well.

My understanding is that it depends on the opponent's move against our move. Am I right?

Almost true but not absolutely. The Score is a mathematical evaluation like for example during the start of the Game it is 0.0. It means both are equal at this point. The Score depends upon a number of factors like more space, pieces, king safety and a lot more.

What is the minimum and maximum range of the score?

The Score is always in favor of Black or White. You can see that when it has a winning combination it can be -infinity to +infinity. The higher the score it means it favors a particular side.

When one can achieve the maximum score at very first move?

It does not depend on you except you are playing your opponent at the beginner level and scholar's mate or fool's mate is played on board.