completely gone

It almost always means that you haven't seen or heard from that person in awhile without explanation, they seemed to vanish.

it can also have a metaphorical meaning. if you say someone has "dropped off the face of the earth" you could be expressing that you don't see the person much any more, or the person is isolating them-self and people are having trouble contacting the person

eastern European explorers once thought the world was flat as in one dimensional. it was thought that if you travel in the same direction on planet earth, that one would eventually fall into outerspace, or like the phase in question. one could fall "off the face of the earth" (1 dimensional face that is)