The web API's do not work with ArcMap directly. ArcMap can be used to author the map resources you host on ArcGIS Server (for which, you will need ArcGIS Server license).
The ArcGIS API for Microsoft Silverlight/WPF enables you to create rich internet and desktop applications that utilize the powerful mapping, geocoding, and geoprocessing capabilities provided by ArcGIS Server and Bing�?� services.