Web browsers use a DNS server to translate the name of the website into the IP of the website. Other programs just connect using the IP.
In the past my DNS server went down, and I could only get places by typing the IP into the address bar instead of the name of the website.
Open the Command Prompt and type "ping www.(pick a site).com", and if it can get to the DNS server it will say "pinging 203.453.23.2343 (ex.)"
Then you know the dns server is working. If not, you can always type your own in, or just wait till it gets ironed out. I ended up calling a friend on a different ISP and borrowing his DNS server.
Edit- if you can obtain an IP address from pinging the name of a website, try entering the IP address into IE and see if it can connect then.