Short answers, pulled from the story.
A sketch is an informal document prepared by a composer to assist in the process of composition. These documents range from small snippets to full drafts and include every imaginable state between unaccompanied melodic motifs and thoroughly worked-out scores.
H. C. Koch and J. G. Sulzer suggested that composers should prepare sketches to lay out how various themes would be arranged to create overall structure within sonata form. This notational form allows a composer like Ludwig van Beethoven to fit together fragmentary ideas made earlier into a coherent whole known as a continuity draft.
The study of composers' sketches has been a branch of musicology since the mid-19th century. Nicholas Marston lists three reasons why this study can be of interest including biographical accounts, determining if a work was created quickly or over years, and showing the workings of a composer's mind.
Surviving sketches by Johann Sebastian Bach include O Traurigkeit o herzeleid which appears as a fragment of a chorale prelude in the Orgelbüchlein manuscript. BWV Anh 2 outlines a cantata for Trinity XIX containing only six bars while BWV 149/1a represents an instrumental opening movement breaking off after the first word with which singers enter.
Watermark studies by Alan Tyson yielded the conclusion that Mozart would sometimes leave a work only partially complete for a number of years before finishing it when an opportunity for performance arose. This practice supports the view that Mozart carefully retained his sketches simply as good business practice to keep open the possibility of future performances and publication.