What is the universal force that causes every object to attract every other object?

Gravity or gravitation.

The larger and (most importantly) more dense the object - the higher (or greater) the gravity.

So gravity is a function of mass (which takes both density and size into account - since something which is very large but has low density) can have a much lower mass than something which is tiny but has a very high density.

Rotation (spinning) and electrical charge can also play a significant role in determing the gravity of any object.