DNS is the "phone book" of the Internet. When you try to connect to a web site, you type in a name such as yahoo.com
The website, however, is actually reached by its IP address. Your computer asks the DNS server for the IP address of Yahoo.com. Normally, the DNS is provided by your ISP, but you can choose others. The fact that you aren't reaching the DNS server tells me that most likely, your router isn't talking to the modem properly. There is also a possibility that you need to contact your ISP support and give them the MAC address of the router so that it will be authorized for their network. Some providers do this, others don't.
Power off your modem and router. Verify that the cable going from the router to the modem is securely plugged in. It should have clicked when you plugged in the cable at both ends. Plug in the modem and wait for 1 min. Once the lights have stabalized, plug in/turn on the router. Your modem should have a link light and the router should have a WAN/internet light.