When main screenwriter John Milius saw a hippie wearing a button-pin with the words "Nirvana Now," he decided to title his film Apocalypse Now. Nirvana is supposed to be final goal of Buddhism, a state of bliss, free from suffering. Instead, Milius' title implies total destruction and death—the end of the world. (Source)
The meaning of the Greek word "apocalypse" is "unveiling of the truth." And since Willard learns the truth about war and evil as he journeys up the river, that could be another dimension to the title's meaning.
We don't actually hear the title spoken; we just see it as graffiti scrawled on the wall of Kurtz's compound: "Our motto: Apocalypse Now." Who wrote it there? Somebody who spoke English, so maybe Kurtz, the photojournalist, or Colby (the assassin who decided to follow Kurtz).
It makes sense that Kurtz himself might've written it, since he's embraced total destruction as the meaning of his own work. Willard sees that Kurtz has written in a manuscript, "Drop the bomb! Exterminate them all!" He knows an apocalypse when he sees one.