How do I install the Cisco VPN client?

You need to install network-manager-vpnc, then configure it through network manager.

sudo apt-get install network-manager-vpnc

That will install vpnc, network-manager-vpnc and network-manager-vpnc-gnome.

Now LEFT click on your network manager applet :

And choose "Configure VPN...", then click "Add" and drop the menu to "Cisco Compatible VPN".

And enter your details. To start your VPN, just follow the steps again, but instead of "Configure VPN", you'll click on the entry you created.

There seems to be a problem on 16.04 with Cisco VPN when installing only network-manager-vpnc. Option for Cisco VPN simply doesn't show in the list.

Another solution is to install

sudo apt-get install openconnect network-manager-openconnect-gnome

then restart network manager. Unfortunately I am unable to provide auth details.

Last but not least is to do this by manually adding file to /etc/vpnc and running it through vpnc command, this one should work, more about it here: https://www.debuntu.org/how-to-connect-to-a-cisco-vpn-with-vpnc/

edit: I probably missed installing something, this works for GUI version:

sudo apt-get install network-manager-vpnc network-manager-vpnc-gnome

Btw, as a performance tip after successfully installing following Scaine's answer, you might also want to enable the option to only use the VPN connection for resources on its network.