Consider the following pictures:
I would like to understand the following:
(Sorry for asking very basic questions. Newbie for the Chess stuff).
Those aren't scores; there are just three possible scores in chess: White wins, Black wins, and draw.
Those are evaluations of the chess engine; a score of +0.62 means the engine thinks White's position is better by 0.62 pawns. A negative score would mean Black is better. On professional level, a score of 1.5 pawns is in most positions decisive, i.e. enough to win the game. However, weak moves by the players will change the evaluation.
The maximum and minimum score depends on the chess engine you're using. If you play the game till checkmate, you will see that the scores will get larger (either more positive or more negative) until you see something like +#8 which means White can mate in 8 moves.
As pointed out by Glorfindel, these are evaluations, which tell you who the engine thinks is better. A positive score means white is better, a negative black is better and values around zero mean that both sides are equal. These evaluation values are (traditionally) displayed with two decimals which is much too high precision for humans. In practical play you would hardly notice a change of +-0.1 or so.
Take these values with a grain of salt, particularly for any evaluation between -1 and 1: there are many cases where the engine evaluations are not good, such as openings or closed positions with lots of maneuvering. Ignore the evaluation in these cases. On the other hand, in very tactical positions you can usually trust engine evaluations. Starting from values above around 1.5 (or below -1.5) in many positions the game ends decisive with a win for white (black) assuming perfect play.
The actual evaluation is calculated based on many factors. The main one is material, and this is also how the evaluation is normalized, namely such that a value of 1 corresponds to one pawn. Similarly other pieces have values of around 3 (bishop, knight), 5 (rook) and 9 (queen). Other factors include piece activity, king safety, control of the center, etc.
With regard to your questions:
What is the minimum and maximum range of the score?
In typical games you will find a maximum of maybe around 20 (or minimum of -20). Beyond that the engine can usually count down to mate and would display the number of moves to mate. If you wish you can probably construct a very closed position (e.g. all pawns blocking each other) and give one player a huge material advantage and in that case you might be able to reach very large evaluation values where the mate is beyond the calculation power ("horizon") of the engine.
When one can achieve the maximum score at very first move?
There is no meaningful answer to this. If there is a first move which gives you maximum score this will depend on the details of the engine and the calculation depth more than on anything else. Another engine might very well give you another best first move. Engines are not powerful enough (or chess is too complex) to give you a good answer to this. That's why engines use opening books and do not calculate anything at the start of the game. Experience has shown that all normal moves (1. e4,d4,c4, Nf3...) give white a small advantage but nothing more.
What is the use of the score?
For the engine, it decides what move to play based on this score. Very simplified you could say that the engine calculates all possible lines down to a certain number of moves and evaluates the final positions using this score. It then picks the line which gives it the best score in the final position.
For a human, you can use this score to analyze your own games or to analyze openings/endgames/... in order to find the correct moves or to locate blunders.
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.
