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: 
This is a consequence of equilibrium.

To see why we must first understand where the normal force comes from. The floor is made up of atoms and molecules and when no one is touching the floor all these atoms are in a 'happy' arrangement. With happy I mean that if you were to move one of these atoms by a small amount it would want to return to its original position. If a man now walks on top of these atoms they all get compressed a little. They are uncomfortable because they are too close to their neighbouring atoms. So they produce a force to try and restore the balance. The more they are compressed the larger the force they produce.

So imagine you hover an 80 kilo man above the floor attached to ropes. He is just touching the floor but is still applying zero weight. Before you cut the ropes the floor is producing zero force; the atoms aren't compressed so they are producing no force. As soon as you cut the rope the man starts falling because of gravity and starts compressing the atoms in the floor. The normal force starts increasing until it matches the weight of the man. At that point the net force is zero and the man is stationary.

Actually the normal force will not match the weight immediately. The man will have also picked up some downwards velocity which needs to be cancelled before he reaches equilibrium. But the end result is clear: in the end the man is stationary so the weight *must* match the normal force.

Before equilibrium is reached the normal force can actually be bigger than the weight. The reason that a ball bounces upwards after you throw it on the floor is because the normal force exceeded the weight of the ball for some time resulting in upwards acceleration.
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.