The first thing to try is of course reboot your phone and your router. Unless you manually alter them, the DNS addresses that get used on your home network are provided by your ISP so rebooting your phone just resets all on your phone but maybe more importantly rebooting your router will reset any glitches that might be the source problem.
If you're still having a connectivity problem, try going into the Application manager in your phone's Settings and find your browser app listing. Open it and tap 'Force stop' and then 'Clear cache'. If your browser continues to show that DNS error message, try installing a DNS changing app like this one:
https://play.google.com/store/apps/details?id=com.burakgon.dnschanger&hl=en
Apps like this allow you to alter the DNS addresses that your phone uses, switching from the ISP supplied ones to a public DNS.
If you're still having this problem after all that, try installing this Netalyzr app:
https://play.google.com/store/apps/details?id=edu.berkeley.icsi.netalyzr.android&hl=en
You use it scan your home network and it will do a thorough analysis, hopefully in the results there will be some items flagged that point out what might be going on.