One may say an orbiter orbits around a planet. But an orbit doesn't have to be around a planet. What is a more general name for the object which is orbited?
The orbiter/satellite doesn't orbit around the planet, it orbits around the center of mass (which, in case of a large difference between the bodies, is well inside the more massive body).
That said, the two bodies in such an orbit are often called 'primary' and 'secondary', see e.g. Wikipedia. You can use it both as a noun (just 'primary') and an adjective ('primary body').
There isn't really an "opposite" in the true sense. Everything which is orbiting or being orbited could be described as an "astronomical object", "celestial object", or "heavenly body".
In our solar system. the Moon orbits the Earth - but in turn, the Earth orbits the Sun. Also, the Sun and all the stars in our galaxy are orbiting the galactic central point (although the latter is not an "object").
An interesting and relevant fact is that a moon is "a moon", not because of its size, but the fact it is orbiting another body classed as a planet. Jupiter's moon Ganymede is actually bigger than the planet Mercury, and if it was orbiting the Sun it would be classed as a planet. One planet has been found that does not have a star. Further, although we haven't found an example yet, it is theoretically possible for a moon to have a moon of its own! So assumptions cannot be made from the terms sun, moon, or planet that it is orbiting anything or being orbited (except for "moon" which must orbit something).
In the case of planets, if you spoke of "Jupiter and its moons" it would be understood that Jupiter is being orbited by other objects. Likewise, if you spoke about a "solar system", this refers to a sun/star which has other bodies (planets, dwarf planets etc) orbiting it.
Really then, there is no need for a word to describe something which is being orbited when the context and terminology make it clear which objects are orbiters.