There are many different ways to write chess moves. The most popular method (described below) is called algebraic notation. The main idea is that every square has a name like this:
                B L A C K
+----+----+----+----+----+----+----+----+
| a8 |#b8#| c8 |#d8#| e8 |#f8#| g8 |#h8#|
+----+----+----+----+----+----+----+----+  ^
|#a7#| b7 |#c7#| d7 |#e7#| f7 |#g7#| h7 |  |
+----+----+----+----+----+----+----+----+
| a6 |#b6#| c6 |#d6#| e6 |#f6#| g6 |#h6#|  F
+----+----+----+----+----+----+----+----+
|#a5#| b5 |#c5#| d5 |#e5#| f5 |#g5#| h5 |  I
+----+----+----+----+----+----+----+----+
| a4 |#b4#| c4 |#d4#| e4 |#f4#| g4 |#h4#|  L
+----+----+----+----+----+----+----+----+
|#a3#| b3 |#c3#| d3 |#e3#| f3 |#g3#| h3 |  E
+----+----+----+----+----+----+----+----+
| a2 |#b2#| c2 |#d2#| e2 |#f2#| g2 |#h2#|  |
+----+----+----+----+----+----+----+----+  v
|#a1#| b1 |#c1#| d1 |#e1#| f1 |#g1#| h1 |
+----+----+----+----+----+----+----+----+
                W H I T E
              <-- RANKS -->
         # # indicate black square

It looks very complicated but it’s not! Each square on a chessboard has a first name (a letter) and a last name a number). The letter identifies a file (column) and the number identifies a rank (row).

Now, the important part is to know how to identify the pieces you are moving. You just need to use a single capital letter:

K = King Q = Queen R = Rook B= Bishop N = Knight

Note that "P" is not used. Chess players have agreed that a move without a letter — such as e4 — is understood to be a pawn move.

You are almost ready to start using chess notation. Just pay attention to the following symbols:

x = Take or Capture 
0-0 = K-side castle 
0-0-0 = Q-side castle 
+ = check 
# = checkmate 
! = good move 
? = bad move 
?? = blunder 

Let’s put in practice what you are learning. We will use the famous "Scholar's Mate", also called the four– move checkmate.

                B L A C K
+----+----+----+----+----+----+----+----+
| R  |#N #| B  |#Q #| K  |#B #| N  |#R #|
+----+----+----+----+----+----+----+----+  
|#p #| p  |#p #| p  |#p #| p  |#p #| p  |  
+----+----+----+----+----+----+----+----+
|    |#  #|    |#  #|    |#  #|    |#  #|  
+----+----+----+----+----+----+----+----+
|#  #|    |#  #|    |#  #|    |#  #|    |  
+----+----+----+----+----+----+----+----+
|    |#  #|    |#  #|    |#  #|    |#  #|  
+----+----+----+----+----+----+----+----+
|#  #|    |#  #|    |#  #|    |#  #|    |  
+----+----+----+----+----+----+----+----+
| p  |#p #| p  |#p #| p  |#p #| p  |#p #|  
+----+----+----+----+----+----+----+----+  
|#R #| N  |#B #| Q  |#K #| B  |#N #| R  |
+----+----+----+----+----+----+----+----+
                W H I T E

1. e4  e5
2. Bc4 h6?
3. Qh5 a5??
4. Qxf7#

See also: algebraic notation

Assimilated from
http://www.uschess.org/beginners/read/

sabre23t has described short algebraic chess notation, the shortened form of long algebraic chess notation, both of which are different from descriptive notation.

Short algebraic notation is now the commonest form of chess notation, probably because it is easiest to understand and looks more elegant.

One thing sabre23t left out was that the move number is an essential part of the notation. For example, white's move in move 4 in the game above would be 4. Qxf7++ (++ is checkmate, btw, there is no notation for double-check). To indicate Black's third move by itself, use three dots, as in 3. ... a5??

The use of exclamation marks and question marks is more for diagnostic and teaching value than anything else.

A single exclamation mark (!) denotes a very good move while a single question mark (?) indicates a blunder. A double exclamation mark (!!) is a rarely used symbol denoting truly brilliant chess moves (ala. Bobby Fischer) while a double question mark (??) indicates a gross blunder. A question mark followed by an exclamation mark (?!) is a move that is questionable but not an outright error while an exclamation mark followed by a question mark (!?) is a surprising move that appears risky.

One more thing - "e.p." is often used to indicate an en passant capture.

Descriptive notation is the old way of writing out chess games. In algebraic notation each square has one name but with descriptive they have two: one from white's point of view and the other from black's. This can make it kind of confusing. Black's perspective is in lower case only to make it easier to read.

              

     Descriptive Notation   

                       Black

 -------------------------------------------------  
 | QR8 | QN8 | QB8 | Q8  | K8  | KB8 | KN8 | KR8 |  
 | qr1 | qn1 | qb1 | q1  | k1  | kb1 | kn1 | kr1 |  
 -------------------------------------------------    
 | QR7 | QN7 | QB7 | Q7  | K7  | KB7 | KN7 | KR7 |  
 | qr2 | qn2 | qb2 | q2  | k2  | kb2 | kn2 | kr2 |  
 -------------------------------------------------  
 | QR6 | QN6 | QB6 | Q6  | K6  | KB6 | KN6 | KR6 |  
 | qr3 | qn3 | qb3 | q3  | k3  | kb3 | kn3 | kr3 |  
 -------------------------------------------------  
 | QR5 | QN5 | QB5 | Q5  | K5  | KB5 | KN5 | KR5 |  
 | qr4 | qn4 | qb4 | q4  | k4  | kb4 | kn4 | kr4 |  
 -------------------------------------------------  
 | QR4 | QN4 | QB4 | Q4  | K4  | KB4 | KN4 | KR4 |  
 | qr5 | qn5 | qb5 | q5  | k5  | kb5 | kn5 | kr5 |  
 -------------------------------------------------  
 | QR3 | QN3 | QB3 | Q3  | K3  | KB3 | KN3 | KR3 |  
 | qr6 | qn6 | qb6 | q6  | k6  | kb6 | kn6 | kr6 |  
 -------------------------------------------------  
 | QR2 | QN2 | QB2 | Q2  | K2  | KB2 | KN2 | KR2 |  
 | qr7 | qn7 | qb7 | q7  | k7  | kb7 | kn7 | kr7 |  
 -------------------------------------------------  
 | QR1 | QN1 | QB1 | Q1  | K1  | KB1 | KN1 | KR1 |  
 | qr8 | qn8 | qb8 | q8  | k8  | kb8 | kn8 | kr8 |  
 -------------------------------------------------  
                                                    
                       White 

The pieces are written as follows,

K = King    Q = Queen    R = Rook    B = Bishop    
N = Knight  P = Pawn

Descriptive is written out piece name first, a dash and then the square the peice is moving to. For the Scholar's Mate descriptive would look like this.

1. P-K4, P-K4
2. B-QB4, P-KR3
3. Q-KR5, P-QR4
4. Q-KB7 mate

Moves are written out in the easiest way possible as long as only one move is legal. For instance, if you're moving a bishop you don’t have to explain which bishop you are moving since there is only one bishop that can be moved. If two rooks can go to Q6, one being on Q8 and the other on Q2, you would write it out R/2-Q6 or R/8-Q6.

Kills are denoted with an "x", for instance NxQ, for a knight killing the queen. If two knights can kill the queen it follows the same rules as above, N/2xQ. Checks are written "ch" and checkmate is "mate". Castling is the same as algebraic 0-0 or 0-0-0 for a queen side castle. When queening a pawn a bracket is used, P-Q8(Q).

This system of notation is still found in the classic chess books but fell out of favor because it was more confusing than algebraic.

In addition to writing down an entire game move by move, there is also a standard method of printing a particular board configuration (fortunately, only one).

The rules are simple:

  • White is at the bottom of the board, Black at the top
  • Each type of piece is denoted by a single letter (P=pawn, N=knight, B=bishop, R=rook, Q=queen, K=king). Although speakers of other languages have different initials for the piece names (and despite the worldwide governing organization (FIDE) being French), I believe that "official" diagrams use the English letters.
  • White's pieces are denoted by upper-case letters, Black's by lower-case

Thus, the original setup of the board is:

+---+---+---+---+---+---+---+---+
| r | n | b | q | k | b | n | r |
+---+---+---+---+---+---+---+---+
| p | p | p | p | p | p | p | p |
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   |   |
+---+---+---+---+---+---+---+---+
| P | P | P | P | P | P | P | P |
+---+---+---+---+---+---+---+---+
| R | N | B | Q | K | B | N | R |
+---+---+---+---+---+---+---+---+

A more complete description of currently used symbols in algebraic chess notation is given below. These symbols are often placed at the end of a move or sequence of moves.

  • = the position is even
  • ∞ the position is unclear
  • += White stands slightly better
  • =+ Black stands slightly better
  • +- White is better
  • -+ Black is better
  • ! a good move
  • !! an excellent move
  • ? an error
  • ?? a gross blunder
  • !? interesting, appears to be good
  • ?! risky, appears to be dubious
  • + check
  • 0-0 castle kingside
  • 0-0-0 castle queenside
  • ++ or # checkmate

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