Weary::Adapter is what allows a Request to occur. By default, Requests occur through Net::HTTP via Weary::Adapter::NetHttp. Weary, as of v1.1.0, provides adapters for Typhoeus and Excon (thanks to contributer Matt Wilson).
An Adapter is really (like other parts of Weary) a specialized Rack application. It doesn't interact with the Weary::Request directly. It instead provides a Rack interface (a method named call with a particular signature). Weary provides a Module (Weary::Adapter) that acts as a simple interface that can be included.