When assembled, Michell's apparatus looked like nothing so much as an eighteenth-century version of a Nautilus weight-training machine. It incorporated weights, counterweights, pendulums, shafts, and torsion wires.
At the heart of the machine were two 350-pound lead balls, which were suspended beside two smaller spheres. The idea was to measure the gravitational deflection of the smaller spheres by the larger ones, which would allow the first measurement of the elusive force known as the gravitational constant, and from which the weight (strictly speaking, the mass) of the Earth could be deduced.
To a physicist, mass and weight are two quite different things. Your mass stays the same wherever you go, but your weight varies depending on how far you are from the center of some other massive object like a planet. Travel to the Moon and you will be much lighter but no less massive. On Earth, for all practical purposes, mass and weight are the same and so the terms can be treated as synonymous. at least outside the classroom.
Because gravity holds planets in orbit and makes falling objects land with a bang, we tend to think of it as a powerful force, but it is not really. It is only powerful in a kind of collective sense, when one massive object, like the Sun, holds on to another massive object, like the Earth. At an elemental level gravity is extraordinarily unrobust. Each time you pick up a book from a table or a dime from the floor you effortlessly overcome the combined gravitational exertion of an entire planet. What Cavendish was trying to do was measure gravity at this extremely featherweight level.