There are two different kinds of weights, when talking bats. Dead Weight and Swing Weight.
The Dead weight is the standard weight if you were to put the bat on a scale and tells you how hard it is to lift a bat.
The swing weight measures how hard it is to swing a bat.
The measure of this resistance is called Moment of Inertia (MOI). The MOI considers where the weight is placed in the bat, not just the amount of weight.
Think of the hammer example. When the head of the hammer is far from your hands it is harder to swing than if you hold the head of the hammer. The further the hammer head is from your hand, the harder it is to swing the hammer.
This is the same with a bat. The lower the MOI, the faster the swing speed.