Question: What do we mean by the term [weight](https://en.wikipedia.org/wiki/Weight)? Is it an upward force exerted by an object to counteract the downward force of gravity (or) is it the downward force itself? Is the magnitude of the weight always equal to the gravitational force regardless of the frame?

Answer: 
It is a consequence of the 3rd law. The man pushes down on the floor, with his weight, and the floor pushes back with an equal and opposite normal force.
The normal force is the perpendicular component of the force that the supporting surface exerts on the resting object. Friction, if there is any is the tangential component. It is a compressive contact force, and is analogous to pushing on a very stiff spring. The harder you (or gravity) pushes on the spring the harder it pushes back. Just like a spring there is also displacement, very small, but non-zero. Unless the surface breaks, it will keep balancing the mg force, so that ma=0.