Final Fantasy's numbering and naming conventions make little sense to an outsider. Hell, they can baffle insiders, too. So IGN asked director Hajime Tabata the deal is with Final Fantasy Type-0, which launches in the west (as a high-definition remaster of the 2011 Japan release) next week.
Basically, it's meant to convey a starting-over for the series.
"We incorporated the meaning of starting anew, from zero," he told IGN. Originally, the game was to be called Final Fantasy Agito 13.
As for naming the characters after playing cards, IGN asked about that, too, see it over at the link. Final Fantasy Agit— uh, Final Fantasy Type-0 launches on PlayStation 4 and Xbox One on Tuesday. For more, see this trailer Square Enix released yesterday. Final Fantasy Type-0 will also deliver a Final Fantasy 15 demo, and for more on that, see our impressions from PAX East 2015.