Wireless router (DSL) connects to your phone line and negotiates with the broadband signal coming in then distributes data to and from its ethernet ports and wireless output.
Wireless router (we'll call this one a cable router) connects to a modem (usually cable but can be DSL) and distributes data to and from ethernet ports and wireless output.
Router can also distribute data between computers which are connected to it.
Wireless adapter is installed in computer chassis (or connected to your computer by USB or PCMCIA) and allows you to connect wirelessly to a wireless router to receive data. Adapter can also be used in ad hoc mode to exchange data between two computers fitted with adapters.