Im applying to an REU in San Diego State where the focus will be Nonunique factorization theory but I'm clueless as to what this actually is. Does anybody know anything about this?

I know nothing about this, but brush up of your knowledge of basic number theory, including primes and factorizations, and you should be fine. This is an REU, so they don't expect you do be an expert, especially on a relatively obscure field such as this. It sounds very interesting. Enjoy!

I have heard of a construction that gives a set of integers that cannot be factorized uniquely. Consider the numbers that have an even number of distinct primes as factors. It is not hard to see that the primes of this set are those integers that have at most two distinct prime factors. Let p and q be two ordinary prime numbers. Then, p^2*q^2=(pq)^2. Since p^2, q^2, and pq have no factors in this set, p^2*q^2 is a counterexample to unique factorization in this set. I suspect that something similar may happen when an even number is replaced with multiples of any integer. Maybe something like this is what you were looking for? You might be looking for something a great deal more advanced. This is just what I've heard that might be related to what you want.

Um, something is happening. I am not reposting this. I think something's going wrong with my computer. I keep clicking edit and i get a new post. SORRY!