Each game in the Final Fantasy series has been stand-alone. They've all had different settings and characters. Some of these games, such as Final Fantasy VII, have branched out into multiple chapters, including prequels and sequels, that share the same cannon setting and some of the same characters. These prequels/sequels can be easily distinguished with their original predessecor by looking at the roman numeral on the title, such as

Final Fantasy VII
Final Fantasy VII: Dirge of Cerebus (non-cannon sequel game)
Final Fantasy VII, Advent Children (cannon sequel game)
Crisis Core: Final Fantasy VII (cannon prequel)

For PSP lovers, and anyone who has played any or all of the Final Fantasy games, Square Enix has presented us with the ultimate Final Fantasy title, bringing together characters from the various different games:

Final Fantasy: Dissidia

http://www.youtube.com/watch?v=oVap1DtAwBQ&feature=related

As you can see by the trailer, it brings together characters from most (maybe all) of the Final Fantasy games, granting us with playable characters such as Sephiroth and many other significant people. It's a 3-D fighting game like Atari's Godzilla games, only with humans and smaller locations rather than monsters with gigantic cities.

I don't know about you, but I'm going to be doing some serious reading up on the other FF games until this comes out, and I'll try to find some clips on Youtube to share with you guys. Check back regulary for updates!

Here's the Wikipedia article.

http://en.wikipedia.org/wiki/Final_Fantasy:_Dissidia

Here is information on the first Final Fantasy game:

http://en.wikipedia.org/wiki/Final_Fantasy_%28video_game%29