The cast is a storehouse for all of the elements of your movie, whether graphics, sounds, quicktime/AVIs or text. (In practice, some of these can be stored outside the Director file, but even then getting them into your movie is usually done via the cast.) Items from the cast are shown in the movie, but the cast itself is not. In most cases, the things appearing onstage are "instances" (that is, copies) of the original castmember rather than the actual cast itself. Multiple instances can be onstage at the same time, even with some of their attributes differing (eg, the same picture at different sizes).
The score is a set of instructions describing how the various cast members crop up in your movie and change and interact over time. The score window represents a kind of "map" of the movie's action in which all the different entities that comprise the content are described.
Director's default behaviour is just to do all the things shown in the score straight through, one after another. This doesn't allow for much in the way of interactivity, so Director also includes a scripting language, Lingo, which can modify the movie's behaviour. Most of the instructions represented by the score can be done explicitly in Lingo instead, and Lingo can also do a lot more besides.
As Director develops, the emphasis seems to be shifting away from the score and towards Lingo as the main site of movie control, but at present the "code" aspect of a movie tends to exist in both places, with the score taking the role of a visual overview. However, it is important to note that, since Lingo can take over control from the score, the picture of your movie which appears in the score window may not be complete.
The stage is where the movie actually plays. Castmembers are drawn and played on the stage according the directions in the score & scripts. The stage is what the user of your movie sees and hears, and where s/he addresses input (clicking on buttons, dragging controls, typing text etc).