I am not sure what specific applications of the adiabatic theorem you are looking at for quantum rings, but I can give you a general overview of the quantum adiabatic theorem and break down some of what those words mean.
As review, remember that energy states and levels in quantum mechanics are represented by eigenstates and eigenvalues, respectively, of your time independent Hamiltonian $\hat{H}$. Thus if we our state $| \psi_E \rangle$ is in an energy eigenstate, then it satisfies the following:
$$
\hat{H}\ | \psi_E \rangle = E\ | \psi_E \rangle
$$
where $E$ would be the energy corresponding to our energy eigenstate.
Now let's say our Hamiltonian now carries some explicit time dependence $\hat{H}(t)$; for example maybe the mass of our particle is now changing in time. In simplest terms, what the quantum adiabatic theorem states is that if your Hamiltonian is changing slowly enough (we'll define this in a second), then if you start in an energy eigenstate $| \psi_E(t = 0)\rangle$, then you will remain in an energy eigenstate $|\psi_E(t)\rangle$ for all time $t$. Thus, you will always have a well defined instantaneous energy for all time (which is what they meant when they said you retain your state and quantum numbers).
If your Hamiltonian is not changing slowly, then in general you will you will have:
$$
|\psi(t)\rangle = \sum c_i |\psi_E(t)\rangle
$$
which means your state is now in a superposition of the instantaneous energy states of your system. I guess in the language of your original question, your state can now "jump" to other simultaneous energy levels, since it will no longer remain in just one.
So how slow is "slow enough"? Sakurai goes through a full derivation to find that the adiabatic approximation holds if the time scale for changes of your Hamiltonian is much larger than the inverse energy of your eigenstate, $\tau \gg \hbar /E$. This is what is meant by "An adiabatic change is one that occurs at a rate much slower than the difference in frequency between the eigen states of energy." This may have been a bit formal, but it gets to the meat behind the quantum adiabatic theorem. Hope this helped.
