Final Fantasy: The Spirits Within is, in the simplest terms, it's a story about life and death.

It is a story about a war.  Humans struggling to survive in a battle against strange and mysterious extraterrestrial creatures, dubbed Phantoms, that have all but destroyed life on Earth.  The aliens don't seem to have a purpose aside from killing anything that lives, and they do that well.

Soon to come, a summerization of the movie and, if I'm feeling particulary motivated, the script.

Square's Official Synopsis As said by the studio itself.

The Timeline A timeline depicting events up to the events in the movie.