Inteet shows connection and Skype works, but sites show a "DNS" error while resolving host

1 down vote favorite I've heard problems like this arise out of DNS errors or something, but as my name suggests, I'm confused on the topic.

I contacted my ISP and they told me to directly connect to modem, which fixed the problem. Thing is, my household needs wifi. The tech support guy told me that the DNS from the modem is the one the router uses, meaning the router is messing something up with the DNS. Is the router faulty?

Please help and I'll explain in further detail things I did and settings I changed etc. Thanks to anyone in advance.

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