Many cutters are designed for specific use and some are general purpose. In many cases the first step in choosing a cutter is identifying the materials that you will be cutting. After you have identified the material there may also be some decisions to make based on the job you are doing and hence the type of cutter you will need.
The action of a saw refers to when the cutting takes place. Most conventional saws cut on the push stroke and some cut on both the push and pull. Japanese style saws cut on the pull stroke. As a general rule push vs. pull is largely a matter of preference with the exception being that saws which can cut on both are typically faster.