Trying to understand what a gravitational wave is
Accelerating masses generate gravitational waves, much like accelerating charges generate electromagnetic waves.
Masses in orbit are continually accelerating, so you're right, Jupiter will generate gravitational waves by orbiting around the Jupiter-Sun centre of mass. In this case there's also gravitational radiation from the sun orbiting around the centre of mass of the Jupiter-Sun system (which is at about the radius of the sun). However the power of the wave emitted is tiny.
Wikipedia gives the following formula for the gravitational radiation of two masses in orbit around each other:
$ P = -\dfrac{32}{5} \dfrac{G^4}{c^5}\dfrac{(m_1m_2)^2(m_1+m_2)}{r^5} $
For the Jupiter-Sun system this gives about 200 Watts - less power than my fridge uses!
For two stars orbiting very closely this power can be much higher - if the two objects are both neutron stars of one solar mass and are separated by 1.9$\times$10$^8$ m, then the power radiated is around 1$\times$10$^{28}$ W, which is quite a bit higher.
Using the inverse square law, the two gravitational wave sources will have the same intensity if their distances are in the ratio 1:10$^{14}$. Since we're about 1.5$\times$10$^{11}$ metres away from the sun, a binary neutron pair's gravitational waves will overwhelm that of the Sun if they're closer than around 10$^{25}$ m away, or about 1 billion light years!
By a wave we normally mean a plane wave.
Possibly this is simpler to understand if you consider electromagnetic waves, as we're all familiar with them. If you're near some arrangement of charges there will be an electric field in your vicinity, and if the charges are moving this field will be time dependent. But we wouldn't normally describe the changing electric field as a wave because it's localised to the charges i.e. if we go far enough away we stop detecting the field. By contrast a plane wave, e.g. a radio wave or a light wave, will in principle travel indefinitely in a vaccum and can be detected at any distance.
The same argument applies to gravitational waves. Near any system of masses there will be a gravitational field that usually changes with time. However this gravitational field isn't propagating i.e. if we go far enough away from the masses we can no longer detect it. However a plane gravitational wave propagates without needing any nearby masses, and will propagate through a vacuum indefinitely.