Files are a through h.
Ranks are 1 through 8.

White starts on the bottom.

Pieces are denoted by:
B = Bishop
Q = Queen
R = Rook
K = King
N = Knight

The idea is to be able to write down the moves of a game so you can replay it later or give it to your teacher for a lesson. When you write down a move, the format is <starting piece>[<source indicator>]<destination square>. If no piece is specified, it means a pawn moved. Sometimes, people use an x to denote a capture. At the end of the game is the result, where 1-0 means White won, 0-1 means black won, and 1/2-1/2 means the game was a draw.

Botvinnik - Kann
Moscow 1931
Dutch Defense

1. d4 e6 2. c4 f5 3. g3 Nf6 4. Bg2 Be7 5. Nf3 d6 6. O-O O-O 7. b3 Qe8
8. Qc2 Qh5 9. Nc3 Nc6 10. Ba3 Bd7 11. d5 Nd8 12. Ne5 de5 13. Be7 Rf7 14. Ba3
ed5 15. Nd5 f4 16. Rad1 Nd5 17. Bd5 Be6 18. Qd3 Bd5 19. Qd5 Nc6 20. Bc5
Re8 21. b4 a6 22. b5 ab5 23. cb5 Na5 24. Qe6 Ra8 25. Rd7 Qg6 26. Qd5 h6 27.
Rc7 Re8 28. Bb4 b6 29. Ba5 ba5 30. b6 Qe6 31. Qe6 Re6 32. Rb1 Re8 33. Rf7
Kf7 34. b7 Rb8 35. gf4 ef4 36. Kg2 Ke6 37. Kf3 g5 38. Ke4 1-0

Move 6 says O-O for White, and O-O for Black. O-O means castles on the King side. O-O-O means castling on the Queen side.

On move 16, white played Rad1. The move is Rd1, but since 2 different Rooks can go to the d1 square, you have to indicate which one. In this case, you use the file, so the “a” indicates the Rook on the a file moved to d1.

Some people and books use other notation symbols. An x in the middle of the move means “capture”, so Black’s move 12 in the game above could be written gxf4, and White’s move 13 could be written Bxe7. A plus sign means check, and it is usually at the end of the move. A pound sign means checkmate. At the end of the game, Black resigned, realizing his opponent would soon start capturing pawns, and eventually make a Queen.