Definitions vary, but typically require the center of mass to be located outside of either object.
In a binary system, the brighter or more massive object is referred to as primary, and the other the secondary.
Visual binaries are two stars separated enough that they can be distinguished through binoculars or a small telescope.
Pluto and its largest moon Charon are sometimes described as a binary system because the barycenter (center of mass) of the two objects is not inside either of them, but Charon is small enough compared to Pluto that it is usually classified as a moon.
The Sun and Jupiter orbit a point outside of either, but are not considered a binary because they are different kinds of objects.