"Between" tells us about a specific relationship. If you are talking about the position of something, for example. The table is between the door and the window -- the table is located in the center, the door on one side, the window on the other side. You can also choose between many options. The emphasis is on making a specific choice, taking one and setting it apart from the many.
"Among" tells us that something is part of a group. Again, if we talk about the location of something, "among" means it is somewhere inside the group. The key is among the junk in the drawer. In this case, we will have to look through the junk to find the key. If we are talking about choosing something, we can say, "That is one choice among many."