Go into the Services window ("Start > Run", type SERVICES.MSC and click OK), then scroll down to the DNS Client entry. Double click it to open its Properties sheet and check that the service is "Started" and that its Startup Type is "Automatic".
Also, it wouldn't do any harm to flush the DNS cache. Open a Command Prompt window ("Start > Run", type CMD and click OK) and, at the prompt, type...
...and press Enter.
I'm assuming that the Wireless Network Adapter in Laptop in question is compatible with the type of encryption set up in the Router (WPA, WPA2 etc)?