Not to be confused with the US release "Final Fantasy III", which is actually Final Fantasy VI
Canonically this is the third Final Fantasy game.
Neither Final Fantasy II or Final Fantasy III were initially released outside Japan, so Final Fantasy IV in North America was released as Final Fantasy II.
Final Fantasy V was also not initially released outside Japan, so Final Fantasy VI in North America was released as Final Fantasy III.
The US Final Fantasy releases were correctly numbered starting with the PSX releases.
From the game box:
A divine balance is disturbed as darkness encroaches on the world, and four chosen youths are entrusted with the final vestiges of hope.
User summary:
Final Fantasy III was the third and last FF installment for the NES. It was originally never released in the US. FFIII features a job-changing system, where the players could revert the player characters into warriors, mages, thieves etc, mid-game. However, the characters needed to stay in one class during a certain number of battle before being able to change again. The dark atmosphere from FFII is replaced with a light fantasy story regarding the restoration of nature's balance through the finding of the four crystals.
Individual credits:
Hironobu Sakaguchi: Director
Nobuo Uematsu: Music
Yoshitaka Amano: Character & monster design
Microbadges: