When we say non-fly-by-wire, or mechanically controlled, we are generally talking about the control surfaces being connected in some way to the control inputs. On older planes, and many newer planes, there is a connection via cables or other mechanical means and the input is augmented with hydraulics, airflow or other methods to make flying the plane less fatiguing.
With some caveats, Boeing uses the above method. Airbus uses fly-by-wire which means the the control inputs are read by a computer and translated into appropriate moment of the control surfaces with zero physical connection. Although time has shown that both methods are incredibly safe, some feel that there can be something "lost in translation" between what the pilot wants the plane to do and what the computer thinks he wants the plane to do.
If there is catastrophic system failure, the Airbus HAS to have some sort of backup power/computer system to operate the plane. With a non fly by wire, theoretically you can have a completely dead plane and still fly it, albeit at times with some great effort.
Anyone feel free to correct me if I got anything wrong.