Weight
Weight is the force of gravity on an object, equal to mass times gravitational acceleration (W = mg) and measured in newtons.
Formula
W = mg
Definition
Weight is how hard gravity pulls on an object. It is a force, so it is measured in newtons, and it changes if you go somewhere with different gravity. Weight is calculated as $W = mg$, where $m$ is mass in kilograms and $g$ is the acceleration due to gravity, $9.8\ \text{m/s}^2$ on Earth, and it is a vector pointing toward the center of the planet; unlike mass, it depends on location. In full, $\vec{W} = m\vec{g}$, where $\vec{g}$ is the local gravitational field with $|\vec{g}| = GM/r^2$ for a spherical body, and because $g$ varies with altitude and latitude, weight is not a fixed property. A scale reads apparent weight, the normal force, which equals $mg$ only when the object is not accelerating vertically. In an elevator accelerating upward at $a$, apparent weight is $m(g + a)$, and in free fall it is zero.
Example
You would weigh about one-sixth as much on the Moon as on Earth, but you would be made of exactly the same amount of stuff. A $50\ \text{kg}$ student weighs $W = 50 \times 9.8 = 490\ \text{N}$ on Earth, and on the Moon, where $g = 1.6\ \text{m/s}^2$, the same student weighs $50 \times 1.6 = 80\ \text{N}$. A $70\ \text{kg}$ person standing on a scale in an elevator accelerating upward at $2\ \text{m/s}^2$ sees the scale read $N = m(g + a) = 70 \times (9.8 + 2) = 826\ \text{N}$, compared to $686\ \text{N}$ at rest. If the cable snapped, the reading would drop to zero.
Key Insight
Mass is how much matter you have; weight is how hard gravity pulls on it, and astronauts in orbit still have all their mass but feel weightless. In everyday life we say "I weigh 50 kilograms," but kilograms measure mass; a bathroom scale actually measures the force you press on it and converts it to kilograms assuming Earth's gravity. Orbiting astronauts are not beyond gravity; they are falling around Earth continuously. "Weightlessness" is the absence of a supporting normal force, not the absence of weight in the $mg$ sense.