There are two ways to use these maps. The first and most simple way is to simply embed the map in an iframe. If you can add HTML content, you can add this:
To centre the map on your area of interest, navigate to your preferred default view on the map and copy the "Permalink" URL in the bottom right of the map. Then replace the URL in the iframe with that URL.
Alternatively, you can use the KML feeds for each layer to embed anywhere you like. They are:
You can download the raw data we are using direct from the OpenStreetMap servers using the eXtended API. Read that wiki page to understand how to use it. To give you a head start, here is a query that will download all the power generators in the Greater London area or the UK:
http://xapi.openstreetmap.org/api/0.6/*[power=generator][bbox=-0.51,51.20,0.35,51.80] (Greater London)
http://xapi.openstreetmap.org/api/0.6/*[power=generator][bbox=-6.5,49.68,2.67,61.31] (UK)
Once you have the data - provided in the OpenStreetMap XML format - you can play with that or convert it into other formats.
You can also download complete dumps of OpenStreetMap data as other formats including Shapefiles. See these pages for shapefiles, SVG graphics and other formats.