Weight

Also called Force of gravity, Gravitational force near a surface

Weight is the gravitational force exerted on an object, equal to mass times the gravitational field strength. It is a vector measured in newtons, and unlike mass it changes if the object moves to another planet.

Weight is a force, so it is measured in newtons and it has a direction: toward the center of the attracting body.

F_g = mg

That equation is derived, not printed. The AP Physics 1 sheet gives you the general law, $|\vec{F}_g| = G\frac{m_1 m_2}{r^2}$ , and the field strength $g = 9.8$ N/kg in its table of constants. $F_g = mg$ is the general law evaluated near Earth's surface, where the distance to Earth's center barely changes over the height of a lab, so the force is effectively constant. The CED treats it as a derived equation for exactly that reason.

It acts at the center of mass. The gravitational force on a system can be treated as exerted at its center of mass, which is why a free-body diagram carries one weight arrow from a single point rather than a fringe of arrows over the whole object.

Weight is not the same as what a scale reads. A scale reads the normal force it exerts on you. Those agree only when your acceleration is zero, which is why you feel heavier in an accelerating elevator while your weight has not changed.

See weight vs normal force for the elevator case, and how to find normal force for the procedure.

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