To protract something is to stretch it out. If you have a disagreement with a friend that you continue for weeks and weeks, you are protracting the argument.
Protract can also mean to draw a plan to scale. Basically when you protract something, you draw it out. You may remember from geometry that a protractor is a device used to draw exact angles. A protractor allows you to protract a plan for the perfect catapult — perfect to use to end a protracted battle.