The Birmingham map can be used in many different ways. You can embed it into your own website or you can download the raw data and create your very own map of Birmingham, for example. There is plenty of information about how to do these things on the OpenStreetMap wiki. The links here provide only some entry points into this documentation:
OpenStreetMap data can be used freely under the terms of the Creative Commons Attribution-ShareAlike 2.0. This means (and this isn't legal advice) basically you can do what you like with the data, so long as you mention the original creator and the licence and anyone else can do the same with anything you produce. For more information you should refer to the legal FAQ.
Here are some examples of applications that people have created with data from OpenStreetMap. Feel inspired!