A published fare is one that is available for purchase by anyone. You could call the airline, or check for prices online, and published fares will be immediately available for purchase.
The rules of such fares are readily available and if there is more than one airline offering the same fare you can count on the rules being virtually the same. A non-refundable fare requiring an advance purchase of 14 days and a minimum Saturday night stay would be just a few of the possible rules of a published fare.
Seat sales launched by airlines are considered published fares as well since (subject to seat availability) such airfares are offered to the public.
Unpublished fares are an entirely different beast. They may be seats that a consolidator purchased and can offer at highly discounted rates. The fare rules could literally contain anything from absolutely no changes allowed to free changes as long as availability exists. They may or may not allow for advance seat selection or the accumulation of frequent flyer miles.
If you called an airline looking for the rules to an unpublished fare you would be out of luck. They are not offered for sale by the airline online or over the phone with the airline.