A weak square is a square that cannot be defended by another pawn.
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Yusupov in Chapter 20 of Build Up Your Chess, The Fundamentals covers Weak Squares and Weak Points. The underlying idea is quite clear. Actually implementing it in a game or position is difficult because there can be many such weak squares, and determining the best targets can be an issue. I don't have the book in front of me but he has a number of examples, and the 12 exercises dealing with them.
The underlying concept is that a weak square is a square in enemy territory that cannot be protected by a pawn and which can be exploited (i.e it is possible to get a piece there). A weak point is a pawn that cannot be protected by another pawn, and that can be exploited (i.e. threatened with capture). If something cannot be reasonably exploited, then it isn't truly weak.
I agree with @madratter7 that Yusupov covers the concept of weak squares quite effectively in chapter 20. Marović also has a chapter on the subject in Secrets of Positional Chess.
This position is from a recent OTB game of mine. I finished the first book of the Yusupov series earlier this year, and the weak square concept jumped into my head in this position.
First realize that pieces of lesser value chase away pieces of greater value (that's why squares that can no longer be influenced by a pawn are important).
Then realize not all pieces can easily influence all squares. Sure it's obvious that a dark square bishop can't influence a light square, but for example it takes a knight on f3 a minimum of 4 moves to land on d5.
So in short, weak squares are squares you have more control of than your opponent. Why would an empty square be important? Because if you place a piece on that square it attacks the surrounding squares. The easiest example is how a rook on the 7th rank can attack and win many pawns.
Hello!!!! Back to weak squares, cluster of them... Any more insights?
The closer to the center the better.
Weaknesses are the pawns or squares on the 5th, and 6th rank which cannot be protected by pawns.
How do you decide on exactly what to attack, when their are multiple weak squares/pawns?
Attack any weaknesses closest to the center first.
I am looking for a book that explains weak squares very clearly. And, gives many good examples .