This is not a very good operational definition of mass, of course. In most common instances, mass is determined by weighing the object and using the force of gravity to calculate the value automatically - which is why you can get on a scale and have read your mass.
The SI unit of mass is the kilogram (kg).
Because of the relationship between weight and mass, these concepts are frequently confused. You can, in fact, convert exactly between weight and mass on the Earth's surface. This confusion is heightened by the fact that in much of metric world, weight is not dealt with, and mass is used in its place almost exclusively. The main difference is that if you were to leave the Earth and go to the Moon, your weight would change but your mass would remain constant.