There are almost as many things to consider when getting a good fit on a running shoe as there are when buying a bicycle. The most basic part of it is the physical size. You want the shoe to be snug enough that your foot doesn't slide around as you run (can you say blisters??) but not so tight that it's uncomfortable. Also, you want to leave some room in the toe box. You don't want your toes hitting against the front of the shoe. And, you want to make sure you have the righ amount of arch support for the shape of your foot.
Some of the other aspects of a good fit depend on the type of runner you are. Do you over/under pronate? Do you have a normal heel strike or are you a toe runner like me? This may affect your decision to get a shoe with more cushioning or with some stability features. Runner's World has some good articles on determining what kind of runner you are and what shoe types are best for you.
I found that it was useful to go to a running store to get fit for my first pair of running shoes. They videotaped me running on a treadmill so they could analyze my running style. They also measured not just the length of my foot but looked at the shape so they could recommend brands/styles that matched up best. Once I narrowed things down to a couple of shoes, they let me run around their parking lot so I could tell what felt best while running.
Beth