This page [adapted from the article ``The mathematical knight'' by N.D.Elkies and R.Stanley in the Math. Intelligencer 25 (2003), #1, 22-34] outlines the ``algebraic notation'' for chess positions and moves, so called because of the coordinate system used to name the squares of the board.
Each square on the 8*8 board is uniquely determined by its row and column, called ``rank'' and ``file'' respectively. The ranks are numbered from 1 to 8, the files named by letters a through h. In the initial array, ranks 1 and 2 are occupied by White's pieces and pawns, ranks 8 and 7 by Black's, both Queens are on the d-file, and both Kings on the e-file. Thus, viewed from White's side of the board (the standard convention for chess diagrams), the ranks are numbered from bottom to top, the files from left to right. We name a square by its column followed by the row; for instance, the White King in the first Diagram below is on d2.
Each of the six kinds of chessmen is referred to by a single letter, usually its initial: K,Q,R,B,P are King, Queen, Rook, Bishop, and Pawn (often lower-case p is seen for Pawn). We cannot use the initial letter for the Knight because K is already the King, so we use its phonetic initial, N for kNight. For instance, Diagram 1 can be described as: White Kd2, Black Ka1, Nf2, Pa2, Pc2.
To notate a chess move we name the piece and its destination square, interpolating ``x'' if the move is a capture. For pawn moves the P is usually suppressed; for pawn captures, it is replaced by the pawn's file. Thus in Diagram 2, Black's pawn moves are notated a2 and axb2 rather than Pa2 and Pxb2. We follow a move by ``+'' if it gives check, by ``#'' if it gives checkmate, and by ``!'' or ``?'' if we regard it as particularly strong or weak. [At some point during this Seminar we'll consider just what a ``strong move'' or ``weak move'' actually means; the notion of a ``strong move'' turns out to be more slippery than one might first think.] As an aid to following the analysis, moves are numbered consecutively, from the start of the game or from the diagram. For instance, in discussing Diagram 1 we'll begin by considering the possibility ``1 Kxc2 Nd3!''. Here ``1'' indicates that these are White's and Black's first moves from the diagram; ``Kxc2'' means that the White King captures the unit on c2; and ``Nd3!'' means that the Black Knight moves to the unoccupied square d3, and that this is regarded as a strong move (the point being that Black prevents 2 Kc1 even at the cost of letting White capture the Knight). When analysis begins with a Black move, we use ``...'' to represent the previous White move; thus ``1...Nd3!'' is the same first Black move.
A few further refinements are needed to deal with promotion and castling, and to ensure that every move is uniquely specified by its notation. For instance, if Black were to move first in Diagram 1 and promoted his c2-pawn to a Queen (giving check), we would write this as 1...c1Q+, or more likely 1...c1Q+?, because we shall see that after 2 Kxc1 White can draw. Short and long castling are notated 0-0 and 0-0-0 respectively. If the piece and destination square do not specify the move uniquely, we also give the departure square's file, rank, or both. An extreme example: Starting from Diagram 3, ``Nb1'' uniquely specifies a move of the c3-Knight. But to move it to d5 we would write ``Ncd5'' (because other Knights on the b- and f-files could also reach d5); to a4, ``N3a4'' (not ``Nca4'' because of the Knight on c5); and to e4, ``Nc3e4'' (why?). [Of course Diagram 3 could not arise in a legal chess game, for at least two reasons. It is the solution to a chess puzzle; can you surmise what the puzzle is, and prove that the solution is unique up to symmetries of the board?]