Are you on Suddenlink? You may be experiencing a problem with your ISP's DNS server. The DNS server is what translates a name like tomshardware.com to 126.96.36.199 so your PC can actually communicate with it over the Internet. When the DNS server goes down it only affects new connection attempts; established connections are still fine.
To test this out, try using Google's free DNS service by going into your router settings, and where it says to use the assigned DNS change that to use these: 188.8.131.52 and 184.108.40.206