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According to this page:

cp 
        the score from the engine's point of view in centipawns.

but is it a score for the current (start) position of the game that I feed to engine, or it is a difference between start and final(when coputations completed and best line found) scores?

And the range in which this value varies is unclear to - how could it be as big as 6000 that I sometimes get for my positions? That would be a ton of figures:)

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It's the score of the current position, but it is computed by static evaluation (i.e. counting pieces, pawns, pawn structure, king safety, mobility, etc.) of positions arising in the program's search tree, and applying the minimax algorithm and alpha-beta pruning.

A cp-score of 6000 does indeed mean the engine thinks White is 60 pawns (centi means 1/100) = between 6 and 7 queens ahead. Because it wasn't able to find a forced mate, it will just try to get the highest possible point advantage, knowing that this will bring the game closer to mate. In practice, a score of (+/-)150 is already decisive most of the times.

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