Royce Simpson
As Derek Law pointed out you can download the template apps and set them up on your web server. It sounds like you already have this part setup and working correct?
Now it sounds like you want to take the configuration settings you've defined in ArcGIS Online and apply them to your downloaded app. The simplest way to do this is to copy the item id for the Basic Viewer app you configured in Online and paste it into the appid field in the Viewer/config/defaults.js file in your locally hosted copy of the template.
Alternatively you can just use the appid as a url param in your locally hosted version. So your app url would look something like this:
http://localhost/Viewer/index.html?appid=718dd403ae7349c285830ac4de9c677b
Once you do this the locally hosted version will read in the app values from the configured online version.
Here are some additional background details on how the templates work - not really important but thought I'd include just in case anyone is interested. There is only one copy of all the configurable app templates hosted in ArcGIS Online. That app has some default values setup and the source code you download is an exact copy of that one hosted app. When you create a new application in ArcGIS Online based on one of the templates a copy of the app source code isn't created we just create an item in ArcGIS Online that stores the values you've specified in the config panel. The template source code knows how to read this online item - which it access via the appid. The item just contains the modifications you've specified in the default panel. Here's an example showing what the info stored in the appid looks like. In this example I've disabled search and modified the theme colors.
If your configured app item is public you can see the values you've set by replacing the id in this url with your own id.
http://www.arcgis.com/sharing/rest/content/items/74791ca2d51345de847dddc52adb4311/data?f=json