Each major chess piece is identified by an uppercase letter, usually the first letter of its name (in whatever language is spoken by the player recording). English-speaking players use K for king, Q for queen, R for rook, B for bishop, and N for knight. (S was also used for the knight in the early days of algebraic chess notation, from the German Springer.)
Pawns are not indicated by a letter, but by the absence of any letter.
Each square of the chessboard is assigned a unique coordinate. The ranks (rows) are numbered from 1 to 8, starting from the row closest to the "white" player. Thus, black's closest rank is rank 8. White's pawns start on rank 2; black's pawns start on rank 7.
The files (columns) are labeled with lowercase letters 'a' through 'h'. The 'a' file is to white's left, and to black's right.
Thus each square of the chessboard is uniquely identified by its file letter and rank number. The square closest to white's left hand is a1; the square closest to black's left hand is h8. The white king, for example, starts the game on square e1. The black knight on b8 might move to a6, c6 or d7.
An ordinary move is represented by the piece's name, followed by the destination square. For example, Be5 would be read as "move a bishop to e5", Nf3 would be read as "move a knight to f3", and c5 would be read as "move a pawn to c5".
Moves with capture are indicated by inserting an x between the piece and its destination. For example, Bxe5 would be read as "a bishop captures the piece on e5". (A colon is sometimes used instead of the x, either in the same place, as in 'B:e5', or after the move, as in 'Be5:'.)
When a pawn makes a capture, the file from which the pawn departed is used in place of naming the piece. For example, exd5 would be read as "a pawn on the e-file captures the piece on d5, or captures en passant so as to end on d5". (It is never necessary to specify that a capture was en passant because a capture from the same file but not en passant would have a different destination square.)
If two identical pieces can move to the same square, additional information must be recorded to remove ambiguity. This is usually resolved by inserting the moving piece's file. For example, with knights on g1 and d2, either of which might move to f3, those moves are Ngf3 and Ndf3.
But if both pieces are on the same file, the moving piece's rank is inserted instead. For example, with knights on g5 and g1, either of which might move to f3, those moves are N5f3 and N1f3.
(In very unusual circumstances, it may be necessary to identify a piece with both its file and its rank. For example, if the player has three queens or three knights on the board.)
The same rules for eliminating ambiguity apply to moves with capture; for example, N5xf3.
Pawn promotion is indicated by appending the piece chosen, for example e1Q or b8B. Sometimes an equals sign is used, for example f8=Q.
Anytime the king moves two squares, it must be castling; so instead of writing O-O or O-O-O, castling may optionally be indicated by recording the king's move; for example, Kg1.
Appending "e.p." to en passant captures is optional, but discouraged. A move which places the opponent's king in double check sometimes has "++" added (rather than "+" or "Ch"). Checkmate can be indicated by appending "#" instead of the word "Mate". (Some use "++" instead, which is discouraged.)
When listing the moves of a game, first the move number is written, then
the move by white, followed by the move by black. For example:
1. e4 e5
2. Nf3 Nc6
3. Bb5 a6
The notation 1-0 at the end of the moves indicates that white won, 0-1 indicates that black won, and ½-½ indicates a draw.