What is my DNS server?

You are using dnsmasq, a lightweight forwarding DNS server that runs locally under the control of NetworkManager. Dnsmasq forwards DNS queries to the DNS servers whose addresses have been provided by the DHCP server. It is also possible to set these DNS server addresses statically using the Connection Editor.

The dnsmasq executable is provided by the dnsmasq-base package on which the network-manager package depends. There is also a package called simply "dnsmasq" which also depends on dnsmasq-base, but this "dnsmasq" package should only be installed if you want to run dnsmasq independently of NetworkManager in order to take advantage of features other than mere DNS forwarding.

As en expansion of happyskeptic's answer, you don't need to specify an interface. This does it for me:

> nmcli dev show | grep DNS
IP4.DNS[1]:                             172.22.216.251
IP6.DNS[1]:                             2a01:4f0:400c:1::1

For Ubuntu 14.04 and older use nmcli dev list | grep DNS

> nmcli dev list | grep DNS
IP4.DNS[1]:                             172.22.216.251
IP6.DNS[1]:                             2a01:4f0:400c:1::1

I've found that on Kubuntu 14.04 the NetworkManager applet (or Plasma thing, whatever it's called) in KDE doesn't show the DNS server info under the connection details.

In this case the way to find it is to run the following at the command line and look for the lines 'IP4.DNS':

nmcli dev list iface wlan0

(replace wlan0 with the interface you use to connect to the Net)