The first question is whether this is a DNS issue, or a connectivity issue. Try to ping 8.8.8.8 (a google dns server). If you get replies, it is just a DNS issue. Try to ping www.google.com. If you get replies, it is a probably a browser issue.
I could answer a lot more accurately with more information, but when you say domain, you do you mean being part of a Microsoft domain, right? If so, are you getting your IP config via DHCP, or is is static? When you are local to a Microsoft domain, the first DNS server MUST be one of your internal DNS servers on the domain, which will, in turn give you DNS records for external domains. Is this a domain you set up at home, or is this at work? Are you able to configure the domain's DNS server yourself?
Lastly, make sure that you have no proxy settings set in your browser.
Could you please post your ipconfig /all (feel free to sanitize the names), but keep the IP addresses, especially for the DNS servers.