The weight of a cat on your lap might not bother you, but the weight of your 156 pound Rottweiler? Oh boy. Weight refers to the heaviness of a person or object.
In physics, weight is a precise measurement based on the force that gravity exerts on a mass. Got that? All meanings of the word weight are related to this sense of heaviness. Something that is important, like testimony at a trial or a serious conversation, can be described as having weight, even if you can't put it on a scale.