You should choose a handsaw based on the type and size of wood you're cutting, and the direction of the cut - either cross cutting (cutting across the grain) or ripping (cutting with the grain).
Saws with fewer teeth per inch provide a faster, but rougher cut and are generally used for "ripping" wood, cutting in the same direction as the grain. The teeth of these rip saws are filed differently than the teeth of a cross cut blade to take advantage of the type of grain in the wood.
Cross cut saws are used for cutting across the grain and have finer teeth, usually from eight to 15 teeth per inch. When cutting thicker pieces, a saw with more teeth per inch may produce more debris than it can handle and possibly clog the cut and slow the cutting process. This can be avoided by using an old candle or paraffin wax and rubbing the blade to make it cut more smoothly through the wood. Remember to hold the saw straight in the cut so it doesn't stick.