No, to the best of my knowledge there is nothing like a general classification of PIDs.  Despite their easy definition, they turn out to be rather a finicky class of rings, as for instance Gauss conjectured that there are infinitely many PIDs among rings of integers of real quadratic fields, but more than $200$ years later we have not been able to prove that there are infinitely many PIDs among rings of integers of *all* number fields.  And, as came out in the comments to Emil's answer, the property of being a PID is not first order, so is not very robust in a model-theoretic sense.  In that regard, the better class of rings are the [Bézout domains][1], i.e., domains in which every finitely generated ideal is principal.  A theorem of Kaplansky which can be used to show that various "big" domains (e.g. $\overline{\mathbb{Z}}$, the ring of all algebraic integers) are Bézout can be found at the end of the section on overrings in [these notes][2].  (I am now giving less precise citations to my often-changing commutative algebra notes in the hope that they will take longer to become obsolete.)

There are some interesting papers on construction of PIDs with various properties.  The one I want to read next is [this 1974 paper of Raymond C. Heitmann][3]: given any countable collection $\mathcal{F}$ of countable fields containing only finitely many fields of any given positive characteristic, Heitmann constructs a countable PID of characteristic $0$ with residue fields precisely the elements of $\mathcal{F}$.  

<b>Added</b>: note that $\overline{\mathbb{Z}}$ is also an **antimatter domain**, i.e., it has no irreducible elements (which specialists in the field tend to call "atoms").  Thus this gives an example of a Bézout domain which is not an ultraproduct of PIDs.  





[1]: http://en.wikipedia.org/wiki/B%C3%A9zout_domain

[2]: http://alpha.math.uga.edu/~pete/integral.pdf

[3]: https://projecteuclid.org/euclid.dmj/euclid.dmj/1077310578