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.
Mass in special relativity
the mass that occurs is the relativistic mass. In other words, the relativistic mass is the proportionality constant between the velocity and the momentum.