Mass is related to weight. It measures the resistance of the substance to acceleration when a net force is applied.
Therefore: armful of lead has more mass
volume measures the amount of 3-dimensional space that an object occupies.
Therefore: kilogram of feather has more volume as it is less dense.
density is the mass per unit volume. Since armful of lead has more mass than a handful of feathers, while kilogram of feather has more volume than a kilogram of lead.
Therefore: kilogram of lead has higher density.
You could let the air out of the balloon while it is under the water with a container filled with water upside down over it. And measure the water displacement.