Zero volts could be any point in the circuit, but to be consistent it is normally the negative terminal of the battery or power supply. You will often see circuit diagrams labelled with 0V as a reminder.

What does it mean when voltage is zero?

Two points in an electric circuit that are connected by an ideal conductor without resistance and not within a changing magnetic field have a voltage of zero. Any two points with the same potential may be connected by a conductor and no current will flow between them.

Is voltage always 0 at ground?

Yes, because there is a potential difference then. Ground can be any voltage you like as long as you state all other voltages relative to ground. So you could call ground 100V and state all the other voltages relative to this if you wanted...