They orbit their common barycenter, as do all objects. Earth and the Moon have a common barycenter which is below the Earth's surface. Pluto & Charon's common barycenter is unique among planets because it is above the surface of Pluto. Several asteroids share this trait, as does the Sun and Jupiter. Thanks to Jupiter, the solar system barycenter is slightly outside the Sun's photosphere, in its corona.
Here's an animated GIF of the Pluto and Charon orbiting their common barycenter. Pluto is purple, Charon is white: