When you transfer momentum between two objects, why is it that the change in total momentum is 0? 

Suppose you are standing motionless on extremely slippery ice. If you now take off your shoe and throw it northward as hard as you can, you will transfer momentum to it. Since you and your shoe were initially motionless, your combined momentum was 0. Neither of you nor the shoe was moving, so the product of mass times velocity was 0. But after you throw the shoe, both you and the shoe have momentum. Your momentum is equal to your mass times your velocity, so your momentum points in the direction you are going. The shoe also has momentum, equal to its mass times its velocity. But since it is heading in the opposite direction from you, it has the opposite momentum from you. Together, your combined momentum remains exactly 0—it didn't change. In general, momentum is transferred from one object to another so that any change in momentum in one object is always compensated for by an opposite change in momentum in the other object.