There are tons of releases, but there's four different ones in terms of mechanics.
The original Super Famicom release that many ports are based off of. The PS1, Wonderswan, GBA, mobile phone (Japan ones, not smartphones) and PSP releases are derivatives of that one.
Then there's the Final Fantasy "II" release in the US. Mechanics were dumbed down, items, spells, abilities and whatnot were dummied out (Cecil's Dark Wave is an example).
There is a third in the form of Final Fantasy IV Easy Type, which was based off the FF"II" release, but restored Cecil's Dark Wave, introduced new items that made things even easier and in some weird cases, enemy patterns were changed. Zeromus has a new sprite (And slightly different attack pattern). The sprite eventually made its way over to become Zeromus EG in the GBA/PSP releases.
Then, there's the DS version which throws everything one would expect out the window. Everything was overhauled basically.
If you want actual releases, here's a list:
Original SFC version
Final Fantasy "II"
Final Fantasy IV EasyType (Japan)
The PS1 release in Chronicles
Wonderswan Color (Which also had very different sounding music! :D)
Game Boy Advance
Nintendo DS
Mobile phones (Japan only. The graphics were redone to match up with the Wii version of After Years, basically)
PSP
iOS/Android (Basically just the DS version)