While so much of modern woodworking is centered around using power tools to make cuts or shape wood, there are times where a hand saw may be a better choice than a circular saw, miter saw, table saw or band saw. This is especially true when a cut needs to be made on a large assembly.
For instance, if you're using doweling as a method of joinery, and after gluing and tapping a section of dowel into the hole to join the two boards (as shown in the photo), a hand saw such as a flexible Japanese backsaw can cut the dowel nearly flush with the surrounding board without causing any damage to the board.
There are two basic types of hand saws: those that cut with the forward motion of the blade, and those that cut with the backward motion of the blade. Traditional hand saws are typically of the forward cutting motion, with a long, relatively stiff blade and a wooden handle behind the blade with the handle nearly perpendicular to the length of the saw blade. These forward cutting saws have two different sub-styles, in rip saw blades and cross-cutting saw blades. Rip saws have a relatively narrow kerf, as all of the cutting teeth are oriented in a straight line down the length of the saw, with deep teeth to clear sawdust from the cut. As the name would indicate, this type of hand saw is used to rip a board, meaning that it is used to cut along the grain of the wood, down the length of the board, just like ripping a board on a table saw.
Conversely, a cross-cut saw blade is designed to cut across the grain of the board. The teeth on a cross-cut blade are not as deep as a rip saw, and the teeth are angled slightly in alternating fashion (one tooth angles slightly left while the adjacent tooth angles the same angle but to the right). This results in a wider kerf than a comparable cut with a rip saw.
These two types of hand saws were commonplace for woodworkers and carpenters, as craftsmen often had a few different saws of each type depending on the cuts required. An experienced carpenter can cut accurate crosscuts nearly as quickly as a woodworker with a miter saw, even though it is a lot more of a physical workout than cutting with a power tool.
The other type of saw is a backcut saw, which cuts on the back stroke. This type of saw is more often used for fine cuts such as the aforementioned cutting off of a segment of dowel after it is glued into the joint. The blades on these saws are thinner than a forward-cutting blade, and much more flexible. The handles of these detail type of saws are most often parallel to the length of the blade. Backsaws really aren't designed for making long cuts, but instead are perfect for detail work. A drywall cutting saw is another example of a back-cutting saw, albeit one with a stiffer blade and a more aggressive tooth.
One great example of the type of detail work for which a back saw is ideal is when cutting through dovetails by hand.
The cuts on the pins or tails of a through dovetail are typically no more than about 3/4 of an inch in length, and must be cut along the edge of the cut line with a backsaw with a very narrow kerf. Once the sides of the pins or tails are cut, the material can be removed with a sharp chisel down to the bottom of the dovetail. In order for the dovetails to fit cleanly together with no ugly gaps, the side cuts must be cut as accurately as possible. This takes some practice, and is one of many reasons why most woodworkers today use a dovetail jig along with a router to accurately cut through dovetails instead of using the traditional hand cutting method.