You Already Know the Basics!
After having just learned how widespread chess is, would it be a surprise to learn that you already know the basics of chess notation? Well, prepare to be surprised, because you already do!
You already know what each square is called, after all, and you know the names of each and every rank and file and diagonal on the board. In addition, you already know the names of all the chess pieces and pawns and the names of their special moves. You also know about check, checkmate, and stalemate.
The chessboard is accompanied by the letters of the files and the numbers of the ranks.
Battleship
Have you ever played Battleship? That's the game where you hide your ships on a grid and try to destroy the ships of your opponent on his grid by calling out coordinates. The grids aren't checkered, but the grid coordinates are none other than chess square coordinates: e1, g2, h4, a7, etc. All right, so there are a few other things to know about chess notation, but again, you are already familiar with everything here.
Recording Your Opinion
You can record your opinion about a move of a chess game very succinctly: Just use punctuation. The following table of punctuation marks that follow moves is understood all over the world. Whenever you see such punctuation after a move, you know that it is the opinion of the annotator (the one writing about the game).
Specifics
Pieces are designated by capital letters: K is for king, Q is for queen, R is for rook, B is for bishop, and N is for knight (not K since that is reserved for the king). The pawn has no symbol. It used to be P, but that has been done away with in the interest of simplicity.
Moves are written as the piece symbol of the piece being moved and the square that piece lands on. Captures are indicated by an x between the piece symbol and the square the capture takes place on. Check is indicated by a + after the move, and checkmate is indicated simply by writing checkmate after the move that produces it. Stalemate is handled in the same way.
Any ambiguities (such as when you have a rook on a1 and a rook on h1 and nothing in between and want to move one of your rooks to d1) are handled by simply adding an extra identifying letter or number. In the case just cited, Rad1 or Rhd1 will convey your meaning precisely.
Keep in mind that all square names and file symbols contain lowercase letters only. Capital letters are reserved for the piece symbols.

Castling is indicated in chess notation by the use of zeros to indicate the number of squares between the king and rook separated by a hyphen. Thus castling kingside looks like 0-0, while castling queenside looks like 0-0-0.
Move Numbers
The moves are all numbered, beginning with the first move, which is move one, and the next move, which is move two, etc. Since White moves first, his move is given first. Next you give Black's move. You just have to remember that Black's first move is part of move one.
Pawn Moves
Just like the movement of the pawn, the notation for pawn moves is different. When a pawn moves, it is written simply as the destination square. You would expect a typical first move to be written 1. Pe4, but actually it is the simpler 1. e4. When pawns capture, the notation is the file letter followed by the x for a capture followed by the destination square (4. dxe5). When a pawn promotes, it is written as the move followed by an equal sign followed by the symbol for the piece the pawn has promoted to (d8=Q or hxg8=N+).
An entire game can thus be described in a single paragraph. Get out a chessboard and set up the pieces for the start of a game and play through the following game.
1. e4 e5 2. Nf3 d6 3. d4 Bg4 4. dxe5 Bxf3 5. Qxf3 dxe5 6. Bc4 Nf6 7. Qb3 Qe7 8. Nc3 c6 9. Bg5 b5 10. Nxb5 cxb5 11. Bxb5+ Nbd7 12. 0-0-0 Rd8 13. Rxd7 Rxd7 14. Rd1 Qe6 15. Bxd7+ Nxd7 16. Qb8+ Nxb8 17. Rd8 checkmate.
This is the final position. If you have come up with something else, go back and make sure you play all the moves correctly.

