Weight
In modern scientific usage, weight and mass are fundamentally different quantities: mass is an intrinsic property of matter, whereas weight is a "force" that results from the action of gravity on matter: it measures how strongly the force of gravity pulls on that matter. However, in most practical everyday situations the word "weight" is used when, strictly, "mass" is meant. For example, most people would say that an object "weighs one kilogram", even though the kilogram is a unit of mass.
Inertia
Gravitational mass is measured by comparing the force of gravity of an unknown mass to the force of gravity of a known mass. This is typically done with some sort of balance. Equal masses will match on a balance because the gravitational field applies to them equally, producing identical weight. This assumption breaks down near supermassive objects such as black holes and neutron stars due to tidal effects. It also breaks down in weightless environments, because no matter what objects are compared, it will yield a balanced reading.