Story is the first mode selectable in a new stage. The stage follows the movie and events play out according to it. You are given a set of predetermined characters and they won't change throughout the level. Not all mini-kit canisters are obtainable because they require other characters that you may not have.
Also of note, in this game only, some stages it is impossible to get the True Jedi status in, I don't know why but this is the case for some levels. I will note them in the main walkthrough.