meaning of numbers (eg+4.4) on game replays and puzzles

Can someone explain (or refer me to a good explanation) of exactly what the score that appears on replayed games means, and how it is calculated?  I have done a search but dont know what the technical name for this running score is and so dont seem to find anything of use....

sure, Chess machines use that to tell who is winning. a score of 0.00 means the game is dead drawn. +1.00 means white is up exactly 1 pawn. -1.23 Means black is up a little more than a pawn. +Mate5 means white has forced mate in 5 moves. -mate5 means black has forced mate in 5 moves

+4.4 means "white's position is equivalent to gaining 4.4 black pawns and so is better than black". -4.4 implied vice versa and black is better.

It's called the position evaluation score of an engine.

i see, thanks.  I tried one of the 'computer workout' problems on chess.com, depending on which move you make, the score changes - it must be a fairly sophisticated algorithm to work out this relative advantage.

One observation however, is that surely the value of 4.4 pawns will vary according to where they are on the board.  3 pawns next to each other on the 2nd row are worse much less than 2 about to promote on the 8th row... 

It's based on the estimated value of pawns early in the game, before position makes a significant difference.

I think future possible continuations are taken into account when determining the worth of a piece. So a white pawn on the seventh row is worth many points if black can't stop it from promoting and getting involved in the game, but it's not worth anything if black can capture it without damaging his/her own position, when white doesn't have a way of stopping it.

The computer doesn't exactly translate everything to pawns, as pawn values might vary. Rather, when each piece was given a base value, pawns were given the base value of 1. Then position is taken into account to add modifiers to either the total value or each piece's value - I'm not sure which. Possibly both.