I’m studying black body radiation and I can’t understand how it produced due to the vibrations of atoms So,in my physics school book it’s written that atoms don’t radiate as long as they are stable ”in the ground state” I can’t understand how I mean isn’t all the bodies radiate as long as they have a certain temperature is all the atoms in the universe are unstable?
 A: 
Ι mean isn’t all the bodies radiate as long as they have a certain temperature

Atoms and single particles in general are not a "body". The "bodies" that radiate are composed out of about $10^{23}$ atoms per mole, and temperature is a thermodynamic variable  defined over a large number of atoms/molecules .
A single atom is a quantum mechanical entity, with electrons in bound states with the positive nucleus, and is stable and will only radiate if it absorbs and reemits a photon, with specific energy allowed by the quantum mechanical solution. An atom  does not have a temperature
Radiation from solid masses of atoms only approximately obeys the black body radiation curve, to the extent that the vibrational and rotational levels of the collection of atoms radiate.
See the approximate black body radiation from the sun measured at the top of the atmosphere, the curve for 5178K (yellow curve)

It has the effect of the spectra of various atoms exciting and de-exiting , plasma interactions too, approximately  fitting  the black body  theoretical formula.
The radiation of the atmosphere at 294K is even a worse fit due to the spectral emissions.


Outgoing spectral radiance at the top of Earth's atmosphere showing the absorption at specific frequencies and the principle absorber. For comparison, the red curve shows the flux from a classic "blackbody" at 294°K (≈21°C ≈ 69.5°F).

A: Atoms in a material are only in their ground states with probability $1$ when $T=0K$, at a non-zero temperature they will have some probability of being in an excited state and so can radiate by spontaneous emission. So in theory if you cooled something to absolute zero it should cease to radiate entirely, but this is only because the atoms constituting the body are stable and don't decay.
If this didn't answer your question please leave a comment.
A: 
Atoms don’t radiate as long as they are stable ”in the ground state”

Yes.

isn’t all the bodies radiate as long as they have a certain temperature

Yes.

is all the atoms in the universe are unstable??

No.
When excited atom emits photons, after some time it will return to a ground state and be ready for next excitation. In ordinary materials most atoms are in ground state at particular time moment. In contrary, most atoms in laser active material are in excited state, due to that it's called population inversion.
Whether or not system is in normal or inverted state, atom behavior does not change,- typical atom juggles between excited and ground states from time to time. Though probabilities of finding n-th atom in excited or ground state can differ as well as probabilities of specific transitions to ground/excited or meta-stable levels.
