Basically, it’s what the name says it is. An authoritative answer comes from a nameserver that is considered authoritative for the domain which it’s returning a record for (one of the nameservers in the list for the domain you did a lookup on), and a non-authoritative answer comes from anywhere else (a nameserver not in the list for the domain you did a lookup on).
It’s basically a distinction between a nameserver that’s an official nameserver for the domain you’re querying, and a nameserver that isn’t. Nameservers that aren’t authoritative are getting their answers second (or third or fourth…) hand – just relaying the information along from somewhere else.
So, for example, If I did an nslookup of maps.google.com right now, I would get a response from one of my configured nameservers. (Either from my ISP, or my domain.) It would come back as non-authoritative because neither my ISP’s nameservers, nor my own are in the list of nameservers for google.com. They aren’t Google’s nameservers, so they’re not the authoritative source that creates the NS records.
The list of authoritative nameservers for Google is below (from whois.internic.net).
Domain Name: GOOGLE.COM
Registrar: MARKMONITOR INC.
Whois Server: whois.markmonitor.com
Name Server: NS1.GOOGLE.COM
Name Server: NS2.GOOGLE.COM
Name Server: NS3.GOOGLE.COM
Name Server: NS4.GOOGLE.COM
Updated Date: 20-jul-2011
Creation Date: 15-sep-1997
Expiration Date: 14-sep-2020
If I changed my configured DNS server to one of the ones in that list, and then did an nslookup against maps.google.com, I’d get an authoritative answer back. Those servers are the authority, (or source) for what are valid names in Google’s domains, and what aren’t. All other nameservers, non-authoritative nameservers, get their NS records from the authoritative servers somewhere down the line.