The original version of "Final Fantasy" was released on the Nintendo Entertainment System in 1990.
The second version in the United States was released for the PlayStation in 2003 as part of the "Final Fantasy Origins" compilation, along with "Final Fantasy II." This version greatly improved the graphics and sound. The script was re-translated and is much more accurate. The magic system now allows you to delete spells you don't want, and the item system was improved to be more like those in later games. Bosses have more HP to make them more difficult. The save system was modified to allow quicksaves and multiple save games. Two FMVs (one at the beginning and one at the end) were added, as were some other special features such as a bestiary and art gallery. It also has an easy mode that can be chosen at the beginning of the game.
The next North American version was released for the Game Boy Advance in 2004 as part of "Final Fantasy I & II: Dawn of Souls." This version is much easier, essentially being equivalent to the easy mode in the PS1 version. The graphics aren't quite as good as the PlayStation version and the FMVs were removed. The magic system was completely changed to be more like the MP system of later games. Some minor changes were made to how characters develop, such as improving the Thief and Monk. You can now save the game anytime outside of battles. The art gallery was taken out, but the bestiary remains. Most significantly, four new dungeons were added to the game. These dungeons, called the Soul of Chaos, are optional, but they allow you to get many more powerful weapons and fight bosses from later games in the series.
The next version was the "Final Fantasy 20th Anniversary Edition," released for the PlayStation Portable in 2007. The gameplay and script was retained from the GBA version. However, it uses the better graphics and music of the PS1 version. It also brought back the art gallery and the FMVs. It still has the four Soul of Chaos dungeons from the GBA version, and it added a new optional dungeon called The Labyrinth of Time.
In 2009, the orignal NES version of the game was released for the Wii Virtual Console. It is more or less identical to the original version. In 2010, the game was released for the iPhone. This version is more or less the same as the PSP version.
Each version has its advantages. People who like retro gaming enjoy the original NES version. People who want better graphics and more of a challenge prefer the PS1 version. If you want the most content and the best translation but don't care about challenge, the PSP version is the way to go. Personally, I would recommend the PSP version.