does the mass of the object have a direct effect on the buoyant force or does the volume? Which of the two has no affect on the buoyant force?

Mass has no effect on the bouyant force. The volume displaced is the cause of bouyant force.

That said, some are sloppy with terms, and say bouyant force when they really mean NET BOUYANT FORCE. Net bouyant force is bouyantforce minus weight, equivalent to additional load that could be carried.