According to Runner's World, the above constitute the three basic foot types, and each has its corresponding shoe. The wet test should help you determine which you are. (The instructions in the video are good, but they do seem a little silly considering how many times you've likely done this on cement after dipping your feet in a swimming pool or something along those lines.)
If you're a flat type, for example, Runner's World says you'll do best with dual-density midsoles and supportive "posts" as well as shoes that offer firmer support. If, on the other hand, you have a high arch—meaning you only see your heel in the test, the ball of your foot, and a thin line on the outside of your foot—you're "best suited to neutral-cushioned shoes because they need a softer midsole to encourage pronation." Those with normal arches have the most leeway and can wear almost any shoe, though Runner's World suggests that "lightweight runners with normal arches may prefer neutral-cushioned shoes without any added support, or even a performance-training shoe that offers some support but less heft, for a faster feel."
Take This Simple Test to Learn Your Foot Type [Runner's World]