Let's see, first, post here the ip address info for your nic. Go to start>run type "cmd" At the command prompt type ipconfig /all and let us know what that says.
When you statically assign an address, make sure it is on the same network as the default gateway(your router) For example, if the router ip is 192.168.1.1, make sure the static ip starts 192.168.1.x then assing x a value between 2-254.
Can your other computer obtain an address via DHCP?
DNS is used all the time. It is only run on servers, but you use it just about everytimr you access a website. When you go to google.com DNS resolves the name "google.com" to the actual ip address. That way we don't have memorize a bunch of ip addresses.
You say you have an internet connection, but can't access a website. Try this:
In a web browser type the following addresses:
http://64.233.187.104
http://68.142.226.51
http://69.93.117.133
Those will take you to google, yahoo and daniweb respectively. If you are connected to the internet, even without DNS, those will work. Let us know the results and we will go from there.