Non-abelian finite groups are quite complicated, but we have completely classified the finite simple groups. Here's an overview from Wikipedia. Moreover, every finite group can be "built out of finite simple groups" in an essentially unique way, in the same sense that every nonzero integer can be written as a product of primes in an essentially unique way! If you want to know precisely what this means, this is the content of the Jordan–Hölder theorem (you will first need to learn what a composition series is). In contrast to prime factorizations, where we just multiply the primes to reproduce the original integer, there are many complicated ways that simple groups can be put together to form a non-simple group. The general problem of understanding the possible isomorphism types of a group from its simple factors is what's known as an "extension problem". In short, while we know all the finite simple groups, and we know that all finite groups are built from these, we are not close to understanding all finite groups.
Non-commutative finite rings are quite complex! Wedderburn's little theorem says that a finite ring with no zero divisors is a field. Besides this, I don't know of any general classification results for finite rings. Even the finite commutative rings are quite difficult to classify: we easily reduce to a product of finite local rings, but those are complicated to describe. The good news is that, by considering the possible underlying abelian groups, it is not too computationally expensive to compute all isomorphism classes of finite commutative local rings of any given cardinality $n$.
I also feel obligated to mention the $5/8$ theorem:
Theorem ($5/8$ Theorem for Finite Groups) Let $G$ be a finite group such that the probability that two randomly chosen elements of $G$ commute is greater than $5/8$. Then $G$ is abelian.
Theorem ($5/8$ Theorem for Finite Rings) Let $G$ be a finite ring such that the probability that two randomly chosen elements of $G$ commute is greater than $5/8$. Then $G$ is commutative.
See here for a proof, or here for a generalization to compact topological groups.