Weight: a gravitational field acting on a mass

Pack a $23\ \text{kg}$ suitcase and fly it to the Moon. It still has $23\ \text{kg}$ of mass: just as hard to swing, and just as painful to catch if someone throws it at you.

Lifting it, though, is suddenly easy. The Moon's gravitational field pulls on each kilogram only about a sixth as hard as the Earth's does. That pull is the suitcase's weight, and it's the one thing that changed.

Near the Earth's surface $g = 9.81\ \text{N kg}^{-1}$, which is exactly the same as $9.81\ \text{m s}^{-2}$ (since $1\ \text{N kg}^{-1} = 1\ \text{kg m s}^{-2}\ \text{kg}^{-1} = 1\ \text{m s}^{-2}$). Every kilogram near the Earth weighs $9.81\ \text{N}$.