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/