Barycenter is the center of mass (i.e., CM) 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 this point to that body's position sum to a zero-length vector. For a system of just two bodies, this simplifies to: mass × distance from the barycenter is the same for each. For some purposes, such as from a long distance away from the orbiting bodies, all of them together can be treated as a single mass at the barycenter, a simplification in calculations. This is the classical-mechanics calculation of the barycenter, but relativity affects its location, and under extreme circumstances, a classical calculation can be significantly off.
A system's barycenter may fall within one of the bodies, which happens when one body is far more massive than the other(s) 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.
While a barycenter is a center of mass, the latter term also applies to a single object: the Earth has a center of mass, undoubtedly very close to a point half-way between the poles, but due to surface topology and density differences within the Earth, is not exactly at that point.
The adjective barycentric means "having to do with the barycenter". In astronomy, 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. (Outside astronomy, the same phrase, barycentric coordinates has a different meaning.)
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 and can be significant in extreme circumstances.