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Technically, weight is the force which an object applies in downward direction, with Newton as its SI unit. However in non-technical usage, weight is a measure of heaviness or lightness of an object, measured mostly in kgs/grams. Is the non-technical definition of weight of an object same as the mass of that object? Or is the non-technical weight also affected by gravity?

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While it is true that sometimes weight and mass are referred to both with the same units (kg, grams etc.) this is technically incorrect (there really is no "technical versus non-technical definitions", but you are right in that the terms are frequently misused).

Mass and weight are two distinct quantities. Mass is a measure of the total amount of matter in a object, and a measure of the degree to which the object will resist a force. It is a scalar quantity. If we let $m$ represent the mass of an object, then the weight it has is given by $${\bf W}=m{\bf g}$$ where $\bf g$ is the strength of the earth's gravitational field. Weight is a force$^1$ and its units are Newtons, and it is a vector quantity.

If you weighed yourself on a scale and got a reading of say $70 kg$, then this is your mass (yes, they are designed to output your mass and not weight by its true definition). Your weight is therefore $$W=70\times 9.8\approx 700 \ \text{Newton}$$


$^1$ To illustrate that weight is indeed a force, note that $$g=\frac{GM}{r_E^2}$$ meaning $$W=mg=\frac{GMm}{r_E^2}$$ which is also Newton's law for gravitational force ($M$ is the mass of the earth, $G$ is the gravitational constant and $r_E$ is earth's radius).

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