The differences between composing, arranging, and transcribing:

 composing: creating musical works
EXAMPLES:

Gustav Mahler
Duke Ellington
Bright Sheng

arranging: reworking existing musical material
EXAMPLES:
Benny Goodman composed "Sing, Sing, Sing"
http://www.youtube.com/watch?v=pW2nkA0vjTU 
William T McDaniel arranged "Sing, Sing, Sing"  for Marching Band. 
http://www.youtube.com/watch?v=QMvQv7pxpxk 


transcribing: transferring music from one medium, or key to another 
EXAMPLES:
J.S. Bach composed "Toccata, Adagio, and Fugue in C Major"  for organ.http://www.youtube.com/watch?v=FilzIfTf8O4 

John P. Paynter transcribed it for the United States Army Field Band.
http://www.youtube.com/watch?v=spP48ntFKOI