Barycenter is the center of mass of two or more bodies orbiting each other, specifically the one point from which the products of the mass of each body times the vector from the point to the body's position sum to a zero-length vector. For some purposes, all the bodies can be treated as a single mass at the barycenter. It may be within one of the bodies, which happens when one body is massive and the orbital radius is sufficiently small. For example, the barycenter of the Earth and Moon is some distance from the center of the Earth toward the moon, yet within the Earth.
The adjective barycentric means having to do with the center of mass. Barycentric coordinates are coordinates using a barycenter as the origin (i.e., a center of mass frame of reference) and can be useful in analyzing orbits.
Classical mechanics uses a simple ratio of masses to locate the barycenter (mass × distance to the barycenter is the same for each of the bodies), but relativity affects its location, so under extreme circumstances, the classical calculation can be significantly off.
The related term, center of gravity (CG) is sometimes used for the center of mass, but more accurately refers to the effect of gravity on an object, i.e., if the object is suspended at that point, the surrounding gravitational field exerts no torgue on the object (force to turn it). If the field is not uniform, an object's center of gravity may not match its center of mass: this happens to an insignificantly-small degree virtually everywhere but can be significant in extreme circumstances.