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Step 1
Notice that the board divided into rows of squares called ranks and files. The ranks are horizontal, while the files are vertical. Since you move vertically to count the ranks and vice versa for the files, this can be a bit tricky, but an easy way to think of this is that the ranks are like lines of latitude and the files are like lines of longitude.
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Step 2
Know that the files are labeled by the letters "a" through "h," while the ranks are labeled by the numbers "1" through "8." Because of this, you can identify each square by its alphanumerical codename. This is really simple. For example, the square in the bottom left of the board is "a1," the one immediately to its right "b2" and the one directly above it "a2."
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Step 3
Learn the abbreviated names of the pieces. Each of the pieces is labeled in algebraic notation by its first letter. Thus, the queen is marked by a "Q," the king by a "K," the rook with an "R" and the knight with an "N," since the "K" was already taken by the king. The pawn, however, is never notated.
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Step 4
Notate a single move by writing down both the piece that was moved and the square to which it was moved. For example, moving a Queen from the bottom left of the board to the top right would be notated as "Qh8." If you had two queens on the board, however, and each of them could reach the h8 square, then you would write the above move as "Qah8," marking the difference between their starting positions.
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Step 5
Notate the capture of a piece by placing an "x" between the two notated piece names. For example, knight on h5 that captures a queen on g3 would look like "Nh5xQg3" in algebraic notation. If the capturing piece were a pawn on h4, though, it would look like "hxg3," since the pawns don't get their own abbreviated name.
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Step 6
Write check as "+" and checkmate as either "++" or "#". The pound sign is more widely used, but both are acceptable and really depend only on your own personal preference. For example, if you checkmate your opponent by moving your pawn from e6 to e7, then write either "e7++" or "e7#."
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Step 7
Know that blunders and brilliancies both have their own symbols ("?" and "!") that are added to the move notation. Forks, skewers and pins usually get at least an "!," but if they're particularly vicious they'll be written as "!!." On the other hand, if the outcome is dubious, they'll be marked with a "!?" or just a "?." It all depends on the strength of the move and the effect it has on your opponent's pieces.
