How to read a game with comments

Chess moves are recorded as coordinates for a refresher look here.  For example 1.e4 e5, 2.Nf3, this allows us to follow recorded moves on a board of our own, or if there are enough diagrams in our mind.  Note: that trying to imaging the moves in your mind is a very useful way to improve at chess!

Many games you see in books, magazines, or on the internet have comments to let you know what the author thinks of a particular move.  Years ago these types of games were difficult to obtain, before the internet information of any kind was difficult to get.  Some publications created a system to let you know if a move was good or not and how the author evaluated the position independent of language, which allowed an annotator to communicate with as many players as possible.  These symbols are still in use today.  

I will briefly explain what each one means and then below explain in more detail.

!    Good move

!!   Brilliant move 

?!   Weak move

?    Mistake

??   Blunder

!?  Interesting move

If a move has no symbol then its considered neutral, or ok i.e. neither that good or that bad.

Good moves

Below are three examples of good moves.  

 Me 33.Rxc5! - After 33...Rxc5 Bxb4 wins back the rook for a two pawn advantage

 Caruana 34.Ne3! - Caruana feels the knight is best placed on e3 and thinks this is a good positional move.

Caruana 12.a6 - Ignores the attack on his own knight with a counter attack

Brilliant

Anand 16...Nde5!! As Black can call check with Qxd4 after the pawn capture by White which leads to mate

Brilliant moves are rare, well perhaps not for the current world champion, but for the rest of us.

Weak

A weak move is a poor move, but not one that is an outright mistake.  The classification of weak moves as a mistake is a matter of taste.  Sometimes it depends on the ability of the players, or the purpose of the game's annotations, often the writer is trying to make a point, so will over emphasise the root cause of future trouble in a game.

NN - 6.Nb5?! is weak because 6...a6 forces the knight to move back losing a move.

Me - 6.Qc1?! is weak as the queen is not well placed on c1.  Note: not a mistake as 6...Qxd4?? would lose to 7.Nb5 as White threatens Qxc8 and Nc7.

Harry - 2... Nf6?!  weak as e5 would force the knight to move straight away.

Mistake

me - 31...Kf5?  Taking the king towards the centre is a mistake, better is Kh7 as Black has his own threats with R2b1 and R8b2 mate.  Black's king is eventually mated in this game.

Szilagyi - 15...c5? a positional mistake as 16.b5 effectively buries Black's dark squared bishop and allows White to use the c4 square for his knight.

Harry - 9.Qe2? A big mistake as White has missed the simple 9.Bxd5 probbaly winning the game.

Blunders
Blunders are the horrible moves where you lose material either directly, or through a simple trick, or you allow a checkmate.

NN - 16.b3?? Allowing Bxc3

Me 42.Qf8+?? which I thought was checkmate missing Bxf8.

Interesting

Alekhine - 8.Nf5!?, interesting because White is giving himself doubled pawns after Bxf5 which may be weak, but Alekhine thinks that he can use the power of his bishops to counter that weakness.  This move may turn out to be good or bad, that is why we are interested to investigate the outcome.