Both the USCF (United States Chess Foundation) and the FIDE (the international governing body for chess) use what's called algebraic notation to dictate moves in chess. In this system, each square has a letter followed by a number. The letters span across the columns, which are called files (such as the d-file). A chess board is 8x8 in size, so valid letters range from a to h. These letters are generally written only in lowercase. Numbers span across the rows, which are called ranks (such as the 7th rank). Valid numbers range from 1 to 8. The 1st rank is always white's side of the board, so when you are playing black, the rank closest to you is the 8th rank. The files a through d refer to the queenside of the board, and files e through h are the kingside of the board. This is true for both white and black (queen is always placed in the d-file, and the king always placed in the e-file).
In order to dictate moves made in chess, you must know which piece you want to move, and the square you want to move to. Each piece has a corresponding capital letter that indicates which piece is being moved. This letter always preceeds the corresponding square. The letters are as follows - K=King, Q=Queen, R=Rook, B=Bishop, N=Knight. The letter P is not used to indicate a pawn. If no letter is written before the square, then it is assumed to be a pawn move. When a knight makes a move, it is often unclear which knight moved, because sometimes both knights may be able to move to the same square. The same is true with rooks. For this reason, whenever the situation with the knight is ambiguous, the original file is generally written before the knight's move, but after the N. If the two knights were originally in the same file, the original rank is written instead. So "Nbd7" indicates the knight that was originally in the b-file has moved to d7, and "R3b5" indicates the rook that was originally in the 3rd rank has moved to b5.
An x stands for a capture. This is written between the piece name and the square. If a pawn makes a capture, it is written with the file that the pawn was originally on followed by an x and then the square. So "exd5" indicates that white has moved the pawn from e4 and made a capture on d5 (we know it must be white because pawns can only go foward, but that is beside the point...). Note that the same notation is used when a pawn moves en passant. Sometimes you might see a (EP) written beside the move to indicate the pawn moved en passant, but generally this is unneccessary. Check is always written as + and checkmate can be written as either ++ or #. Short castling with the rook moving two squares is called kingside castling, and is written as O-O. Long castling with the rook moving three squares is called queenside castling, and is written as O-O-O. Chess moves are often evaluated when they are written. ! indicates a good move, and ? indicates a bad move. Furthermore, a very good move is written as !!, and a very bad move is written as ??. A questionable move is written as ?!, and an interesting move is written as !?.
Lastly, you must understand how moves are grouped together. When writing out moves one by one, you always count the moves, and that number is always written first followed by a period. Then white's move is written, followed by a space, and black's move. Often times you might want to refer to a certain move. Do this but simply writing the number of the move, followed by either white or black's move, depending on which one you're referring to. If you refer to black's move, you must write the number of the move followed by three periods "...". So for an example, "6. ... Qh4" indicates that black moved his queen to h4.