[Coda]

Unfortunately, translating that into English just says "we've got a connection to the router, but we can't communicate with a required Internet service through it." Which you mostly already knew.

If the problem is specifically something to do with DNS (as opposed to that being just a symptom of a different problem) then there are things we can do.

It's possible that your modem has a buggy DNS server. (This is annoyingly common, especially with the cheap crap you get from ISPs -- they can handle a single user but they fall apart as you start putting more load on the Internet connection.) If this is true, then web addresses that you've already looked up will continue working for a while, and connections you've already established will continue to work but new connections will fail to be established. This would produce the message you saw, and it would mean that other people on your network might not notice the outage if they're sticking to just doing the same thing the whole time.

If the problem is indeed with your router's DNS server, then the best thing to do would be to configure your computer to not use it -- which is what I had mentioned earlier. Google's DNS servers are fast, free to use, and have something like 99.999% (maybe even 99.9999%) uptime. Those servers are 8.8.8.8 and 8.8.4.4.

What I said you should look up is instructions for how to set the DNS servers. I don't know what version of Windows you're using, and the specific steps vary, but you shouldn't have any trouble finding what you need.