Correct Answer:
Your DNS settings are incorrect. Given that most systems use DHCP, it's rare to have a problem like this, but if you typed in invalid manual DNS settings into your NIC's Properties, this can happen.
Incorrect Answers:
If you were to turn off the DNS Client service on your computer, you would get this problem. However, there is a much smaller chance that this service would be disabled than that someone had entered an invalid DNS service IP address.
In a single network, everyone USUALLY uses the same DNS server, so if the DNS server is down for you, it's down for everyone else. Granted, you might have a different DNS server, so this answer is valid, but not as valid as entering an invalid DNS server IP address.
Routers don’t have an item in their Access Control Lists called "DNS permissions". You COULD block DNS from getting through your router by blocking port 53 (DNS), but then no one could get DNS access. Most routers have the ability to block a port number for a single IP or MAC address, but, again, it's simply too unlikely.