It looks like avahi-daemon is started when the network connection is established (
This notification is informing you that mDNS (Avahi) has been disabled. It's only used for a small number of applications that only work on the local network, it won't adversely affect your internet connection or DNS.
The most well known use for mDNS is sharing music with Rhythmbox (or iTunes) over your LAN. It's an Apple technology, but it's largely been ignored in favour of uPNP or DLNA.
To disable it, you must edit the file
/etc/default/avahi-daemon as root:
and add this line (or change it if already exists to):