Ubuntu 16.04 LTS, internet connection doesn't work, but network still can access machine

It's my understanding that /etc/resolv.conf is for public nameservers only. It appears that yours has been modified to use a private nameserver ostensibly over a VPN. If you wish your Ubuntu box to obtain Domain Name Service normally to allow internet access, I suggest that you revert back to the default /etc/resolv.conf file which contains:

# Dynamic resolv.conf(5) file for glibc resolver(3) generated by resolvconf(8)
#     DO NOT EDIT THIS FILE BY HAND -- YOUR CHANGES WILL BE OVERWRITTEN
nameserver 127.0.1.1

You can revert back to the default /etc/resolv.conf file by either restoring the original from a recent backup from before it was modified, or opening the file the link points to with the command sudo gedit /etc/resolv.conf and pasting the code from the default file above into it overwriting it's current content. You may want to save a copy of the current content beforehand as a backup somewhere you will be able to locate it if you find you need it.

NOTE: /etc/resolv.conf is a symbolic link to the file /run/resolvconf/resolv.conf so file based operations such as editing on /etc/resolv.conf will actually be carried out on /run/resolvconf/resolv.conf

To confirm that the link hasn't been erroneously modified run the command ls -l /etc/resolv.conf which should result in the output below.

lrwxrwxrwx 1 root root 29 Feb 25 2016 /etc/resolv.conf -> ../run/resolvconf/resolv.conf

the arrow indicates it's a symlink.

For more information on resolvconf which modifies this file see the man page.

If you have an unmodifed /etc/resolv.conf file (and you should) you should change System Settings -> Network -> Options to be set as shown below so that your router will provide DNS services to your Ubuntu box as it does to your other systems.

An alternative is to use Google public DNS service and add 8.8.8.8 and 8.8.4.4 as your DNS servers.