Hello. I like set theory. I will try to explain what I've learned here.
We take axioms to be statements which are true without prior logical justification. From axioms, we can derive new true statements and impose additional restrictions to create structures.
We usually work with Zermelo-Fraenkel with Choice, ZFc. Sets are such that, for two sets A and B, the statement A \in B evaluates to true or false. That is, you can ask whether a set B contains a set A, and you will always get an answer.
See here for a philosophical justification for the axioms. One of the purposes of set theory is to be as broad as possible so you can build off of it for other areas of math.
The first two axioms of ZFC define sets:
1) Extensionality: two sets are equal if they contain the same elements.
2) Foundation: No set contains an infinite descending chain of \in; for any set A, if it contains another set B, and If B contains another set C, and C contains another set D... this will terminate after a finite number of iterations. This implies no set contains itself, as if it did, you'd have an infinite chain. Who cares there's like a bunch o fthem and they all contradict eachother you've got like Choice which says every set has a bijection with an ordinal basically and determinacy which contradicts choice an dsays infinite games have a set strategy and 1 == 0 which is hahah funny