There are several types of saws for cutting wood. Some are used to make straight cuts and others are used to make curved cuts.
The two main types of handsaws in the western world are the crosscut saw and the ripsaw. The teeth on a ripsaw are like a row of chisels, each used to sever a fiber of wood as they pass parallel to the grain. The teeth on a crosscut saw are like a row of knives, each used to slice across the wood fiber as they pass perpendicular to the grain. As its name suggests, a crosscut saw is used to cut wood perpendicular or at an angle to the wood grain. The crosscut saw typically has more teeth per inch than the ripsaw. A crosscut saw can also be used to cut plywood. The ripsaw is only used to cut wood in the same direction as the grain. The ripsaw makes a very rough and ragged cut when used to cut across the grain. The crosscut saw is a better all-purpose saw since you can cut with or against the grain. Although it can tend to follow the wood grain and make it harder to make a straight cut with the grain.
Both of these western handsaws cut on the push stroke. Japanese saws, on the other hand, cut on the pull stroke. The advantage of the Japanese saw is the width of the blade can be much thinner because you are using the steel in tension. Western style saws are much thicker because they will buckle if they are too thin. An exception is the backsaw used by cabinetmakers. This saw has a very thin blade with a high tooth count (for cleaner cuts) and a stiffener across the top of the blade to keep it from buckling.
The compass or keyhole saw is a short, narrow-bladed crosscut saw used for getting into tight spaces or for making circular cuts
The coping or fret saw has a narrow fine-toothed blade held in a c-shaped body and is used to make circular cuts and to for cope (trim) molding to fit snugly.
The hacksaw has a very fine-toothed, narrow blade and is used for cutting through metals.