Conceptually I understand that mass is a measure of the amount of stuff present in an object, while weight is a measure of the force of gravity applied to that object. An object of a given mass will have a bigger weight on Earth than on the moon because Earth’s gravity is stronger. But… mass is determined by weighing an object on a scale. And there is a simple mathematical conversion between grams (mass) and pounds (weight), implying that they’re just different units for the same concept. So what gives? (Also this is a question that applies to so many fields, I had no idea which flair to apply.)
In: 53
The other answers have addressed most of your question, but I wanted to draw attention to:
> implying that they’re just different units for the same concept
While they’re not the same (as others have said), you did hit on something really important here: inertial mass (how much something resists acceleration) and gravitational mass (how susceptible something is to a gravitational field) _are_ always equal, and this is not an obvious thing! For example, there’s no connection between inertial mass and how susceptible something is to an _electromagnetic_ field. We’ve always measured the two kinds of masses (inertial and gravitational) to be equal, but there’s no inherent reason to assume they should be, absent those many observations. Einstein called the assumption that they’re equal the [equivalence principle](https://en.wikipedia.org/wiki/Equivalence_principle), and it’s an important part of general relativity.
tl:dr: your confusion is in large part due to the fact that these two logically-independent definitions of mass are always equal, and that turns out to be a non-obvious yet very important fact about the universe.
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