What is the definition of the "moving part" and "fixed part" of a linear system$|L|$?
I think the fixed part should be defined to be the greatest effective divisor $F$ such that $D-F\geq 0$ for every $D$ in the system, and the moving part is the linear system $|M|=|L|-F$.
Thus the fixed part is the codimension 1 part in the base locus.
$|M|$ may not be point free?(but I think for the curves, it is basepoint free)
If $|M|$ defines a rational map (morphism on some open subset) to $P^k$, what is its relation to the rational map defined by $|L|$?
When people say moving a divisor in the moving part, does it always mean using implicitly the Bertini theorem?
Is there any reference on the moving and fixed part of linear system?
Some quick answers:
-Yes, the $\geq$ version of your definition is correct. (In the definition of moving part you have a typo; replace $D$ by $F$.)
-In general $|M|$ is not basepoint-free. Certainly that is true for curves, because there "fixed part" and "base locus" coincide. For surfaces $|M|$ might not be basepoint-free, but it is a theorem of Zariski that $|kM|$ is for some natural number $k$. For dimension 3 and more nothing like this is true: look up the definition of "movable linear system".
-Any rational map can be extended to codimension 1. In your situation, the rational map defined by $|L|$ will extend to the morphism defined by $|M|$.
-"Moving" just means choosing another effective divisor in the same linear system. No need to refer to Bertini's theorem.