A whole game of chess fits on a line of text: 1.e4 e5 2.Nf3 Nc6 3.Bc4. That shorthand is algebraic notation, the way players everywhere write games down, and every book, video and engine uses it. It is also how every line on this site is written, so it is worth ten minutes.
A move needs two things: which piece moved, and where it went.
Every square has a name
The columns are files, lettered a to h from White's left. The rows are ranks, numbered 1 to 8 from White's side. A square is its file then its rank, so White's bottom-left corner is a1 and Black's is h8. The letters and numbers run along the edge of the board here, and on the trainer's too.
The names never change with the side you play. From Black's chair a1 is in the far right corner, but it is still a1.
Every piece has a letter
Kking,Qqueen,Rrook,BbishopNknight, because K was already taken- Pawns have no letter at all
Other languages use their own letters, and some books print little piece symbols instead, like ♘f3. The squares stay the same everywhere.
A move is a piece and a square
Write the piece's letter, then the square it moved to: Nf3 is a knight moving to f3. A pawn move is just the square, so e4 is a pawn moving to e4. You never need to say where a piece came from, because only one piece of that kind can usually get there.
Moves are numbered in pairs, White's then Black's: 1.e4 e5 is White's first move and Black's reply. Step through these to see each one played.
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When a Black move is written on its own, three dots stand in for White's: 3...Bc5 is Black's third move.
Captures
An x means a capture, between the piece and the square: Qxd5 is the queen taking whatever was on d5. A pawn has no letter, so a pawn capture starts with the file it left: exd5 is the e-pawn taking on d5.
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Check and checkmate
A + after a move means it gives check. A # means checkmate, and the game is over.
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Castling, promotion and en passant
O-Ois castling kingside, the short side.O-O-Ois castling queenside, the long side.- Promotion adds the new piece after an equals sign:
e8=Qis a pawn reaching e8 and becoming a queen. - En passant is written like any pawn capture, to the square the pawn lands on:
exd6. Some writers adde.p.after it.
Each of these is explained in how the pieces move.
When two pieces can reach the square
Sometimes two pieces of the same kind could go to the same square. Then the move adds the file the piece came from, or its rank if the file doesn't settle it: Nbd2 is the knight from the b-file going to d2, and R1e2 the rook from the first rank going to e2.
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! and ?
Writers judge moves with marks after them. They aren't part of the move, just an opinion:
!a good move,!!a brilliant one?a mistake,??a blunder!?interesting, worth a try;?!dubious
Notation on this site
Every line in the openings is written this way, and each one is labelled by the opponent's moves alone. The London's first line, for instance, is “vs 1...d5 2...Nf6 3...e6 4...c5”: Black's first four moves, with White's left for you to find. The trainer's move list writes each move the same way as you play it.