A barycenter is a physics term - or astrophysics, more precisely. If two objects are orbiting each other - or one is orbiting the other, technically - the barycenter is the center of mass of the two bodies. This is also the point around which they orbit. In the case of probably the most familiar pair of objects, the Earth and the Moon, the barycenter of the Earth-Luna system is somewhere approximately 0.75 of Earth's radius measured out from the center of the Earth towards the center of mass of Luna. So it's a boring barycenter, because it's underground beneath our feet.
However, there are much more interesting examples. Alpha Centauri (α Centauri A) and its companion (α Centauri B), for example, are at least a binary star system and possibly a trinary star system, depending on whether Proxima Centauri (α Centauri C) is gravitationally bound to the other two. In this case, the barycenter of the Centauri system is somewhere between Alpha and Beta, possibly slightly biased towards Proxima. In other words, in multiple star systems the barycenter becomes more important. For example, I think the barycenter is similar to a Lagrangian point - in other words, if you could park an object at the barycenter of a multiple star system, it would remain in place and not begin falling towards a member star. This is probably impossible in practical terms however; the barycenter's location would also have to take into account the rest of the mass in the systems, and in addition might move as the stars precess.
Thanks to locke baron for pointing out that Beta Centauri != α Cent B)!