So I know the list of lists of aleph-numbers that goes:
$\aleph_0, \aleph_1, \aleph_2, \aleph_3 ...$
$\aleph_{\aleph_0}, \aleph_{\aleph_1}, \aleph_{\aleph_2}, \aleph_{\aleph_3} ...$
$\aleph_{\aleph_{\aleph_0}}, \aleph_{\aleph_{\aleph_1}}, \aleph_{\aleph_{\aleph_2}}, \aleph_{\aleph_{\aleph_3}} ...$
$\aleph_{\aleph_{\aleph_{\aleph_{\aleph... \aleph_0}}}}$, $\aleph_{\aleph_{\aleph_{\aleph_{\aleph... \aleph_1}}}}$, $\aleph_{\aleph_{\aleph_{\aleph_{\aleph... \aleph_2}}}}$, $\aleph_{\aleph_{\aleph_{\aleph_{\aleph... \aleph_3}}}}$ ...
How do these levels correlate with the ordinals? Is $\aleph_{\aleph_1}$ equal to $\aleph_{\omega_1}$ or $\aleph_{\omega+1}$ or what? Where would $\aleph_{\omega^\omega}$ be? Is there an $\aleph_{\epsilon_0}$? An $\aleph_{\epsilon_{\epsilon_{\epsilon_{\epsilon... \epsilon_0}}}}$? And so on? From my reading it sounds like there "must" be, maybe?
For every ordinal $\alpha$, the cardinal $\aleph_\alpha$ is defined. Since cardinals are themselves ordinals, they fit into this order as well.
For example, since $\aleph_0 = \omega<\omega+1$, we have that $\aleph_{\aleph_0} = \aleph_\omega < \aleph_{\omega+1}$. Similarly, since $\left|\omega^\omega\right|=\aleph_0$, we have $\aleph_{\aleph_0} < \aleph_{\omega^\omega} < \aleph_{\aleph_1}$
Some other important facts:
If $\alpha<\beta$ are ordinals, then $\aleph_\alpha<\aleph_\beta$.
For any cardinal $\kappa$, there is an ordinal $\alpha$ such that $\kappa=\aleph_\alpha$.