Doc:
The FF games are tied together by the same creator, same character/costume designer, and pretty much the same music composer. They have similar combat styles though that is changing with the newer games. Many of the games have been about saving 'crystals' which usually protect the world in some fashion. The vast majority of the games have been about saving the planet (whichever planet that might be in a particular game) from environmental degredation/badguys/extra-planetary beings that want to absorb it.
The Spirits Within captured the essence of that "saving the planet" mentality, but other than that, really had nothing to do with the content of the games- aka random monster battles, dragons, swords, spells, blend of high-tech and low-tech (sword vs. machine gun = even battle), and so on.
As far as the movie goes- FF: Advent Children is a 'sequel' to FF7. It picks up a few years after the game left off and contains all the same main characters with some new additions. It also contains some plot sequences that are very familiar to anyone who has played the game. I imagine it's a fun watch if you know nothing about the game, but fairly devoid of meaning (though don't get me wrong- it's an action flick, not a deep philosophical movie). The official DVD version has a review of the events of FF7 for those who haven't played it so they can understand more of the movie.