I'm not sure what I'm missing here...
Why are there positive and negative ions in the depletion region?
Joe gives a good overview of how a hole is created when an acceptor atom captures an electron that was previously moving freely in the valence band. Similarly a conduction-band electron is created when a donor atom releases an electron into the conduction band.
I am guessing that what you're missing in this is that the acceptor is an impurity atom in the crystal, and it is fixed in location (unless we start talking about the extremely high temperatures that are used during the diffusion process to actually get the impurity atoms into the crystal in the first place). Whereas the hole is free to move around. So the hole is free to be pushed around by the built-in field of the junction, but the acceptor (now a negatively charged ion) is stuck where it is.
I thought there'd be neutrally charge ions.
The word "ion" means an atom with a net positive or negative charge. If it weren't charged one way or the other, we'd just call it an "atom" and not an ion.
So why do the acceptors become ions instead of just staying neutral?
One way to look at it is that the Fermi-Dirac statistics tell us what is the probability of any particular state being filled with an electron at any particular time. Since acceptor atoms, for whatever reason, provide states that are near the valence band edge (and so below the Fermi level) those states are going to end up being filled more than 50% of the time.