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: 
@AccidentalTaylorExpansion gave the accurate answer and the proper reasoning that the **normal force depends on the intermolecular distances**.

You can visualise it actually in a ground with mud.

When you try to walk, your foot goes into it and after a certain depth you may stop sinking. **This shows that when you came in contact with the surface of mud , your weight was not equal to the normal contact force but when you pressed the mud the atoms came closer and after a certain distance it equalised your weight** and this is the point where you came to rest and ready to take the second step.

Hope it helps .
Weight is defined as *[the force with which the body is acting on its support or the suspension](https://en.wikipedia.org/wiki/Weight#Operational_definition)*, whereas the normal force is the force with which the support acts on the body. (Note that this is different from what is understood by *weight* in everyday life - the measure of gravitational mass.)

Depending on a situation these may or may not form a 3rd law couple:

* If a block rests on a horizontal surface, the weight and the normal force form the third law couple
* On an inclined surface, the weight is composed of the force counteracting the normal force and the friction force
* Adding a supsension to either of the above two cases may further complicate the situation.