Division is breaking something down into parts, splitting it up. When you have division of labor, everybody does her part.
Division is a noun. It refers to taking a hunk — of pie, clothes, numbers, people — and splitting it into smaller bits. A division in the military sense is a group large enough to go to battle. Blame British mathematician John Pell for the long division you learned in math class (but proceeded to forget once you discovered the calculator). Any way you slice it, division is the process of separating something into smaller parts.