Material objects are, in general, composed of molecules, of which the building blocks are atoms. The periodic table gives a classification scheme for the kinds of atoms, called elements, which can occur. We have very probably a complete classification of all naturally occurring elements, although it is conceivable that we can make more.
Similarly, integers are built out of primes in a unique factorization. There are infinitely many prime “elements”, but we don’t know just exactly where the next one is going to show up. The prime number theorem does give us a sense of how they look if we squint and peer out over the horizon, and I believe the Riemann hypothesis would provide more clarity if proved.
Finite simple groups are another class of “atoms”, and their classification is a monumental result in algebra that I probably will never understand. Abelian groups, on the other hand, are much easier to classify, even when infinite ones (that are nevertheless finitely generated) are allowed.
My question, then, is this: What are some other “periodic table” results in mathematics? Are there other subfields in which some important class of objects was largely or completely categorized?