In the 2 body problem, if you just set the bodies down, they'll be attracted toward another and then collide. If either one has some velocity component perpendicular to the line between them, there will be one or another type of relationship depending on the eccentricity http://en.wikipedia.org/wiki/Orbital_eccentricity
It might be Parabolic, http://en.wikipedia.org/wiki/Parabolic_trajectory
Hyperbolic, http://en.wikipedia.org/wiki/Hyperbolic_trajectory
or an elliptical orbit, one special type of which is a perfect circle
this page is nice http://en.wikipedia.org/wiki/Two-body_problem
Do Jupiter and Sun orbit their common barycenter? Technically no because of the perturbations of other bodies in the SS, but they would if they were alone.