When I'm connecting broadband using D-Link ADSL broadband Modem, the connected computer will connect to Internet after verifying username and password. If I'll replace that modem with D-Link DSL-2640B (A broadband ADSL Modem WiFi router) then will this router can connect to Internet automaticall when switched on OR A computer is still required to connect to Internet? If DSL-2640B will connect to Internet automatically after switch on then will Internet connection automatically disconnected when switched off? If DSL-2640B will not connect automatically then may I configure it to connect Internet automaticall without using Computer?

A broadband router of the type you describe deals with the connection for you.
Basically the router is always connected so any machines that are linked to the router have internet access on start up.