The Zermelo-Fraenkel axiom system with Choice (ZFC) is the standard foundation for contemporary mathematics. It replaces naive comprehension with a carefully controlled list of nine axioms and axiom schemas: extensionality (sets with the same elements are equal), pairing, union, power set, infinity, separation (restricted comprehension), replacement, regularity, and choice. Together these axioms permit the construction of all standard mathematical objects — the integers, reals, functions, topological spaces — while avoiding known paradoxes. By Gödel's second incompleteness theorem, the consistency of ZFC cannot be proved from within ZFC itself.
Survey all nine axioms before studying any one in depth — categorize which axioms assert existence (pairing, union, power set, infinity), which restrict (separation, regularity), and which assert closure under operations (replacement). Then return to each axiom individually and ask: what can I now build that I could not build before?
You already know that naive set theory runs into contradictions — most strikingly Russell's paradox, where the set of all sets that don't contain themselves both must and cannot contain itself. The project of ZFC is to start over with a short, explicit list of axioms that permit enough set-building to do all of mathematics, while avoiding the unrestricted comprehension that caused the trouble.
The key move is replacing "for any property P, there exists a set of all x satisfying P" (which allows Russell's construction) with the Axiom of Separation: "for any property P and any existing set A, there exists the subset of A satisfying P." You can only carve out subsets of sets you already have — you cannot conjure a set from thin air by specifying a property. The other existence axioms (pairing, union, power set, infinity) tell you what new sets you can build from ones you already have. The Axiom of Replacement says that if you can define a function on a set, the range of that function is also a set. Together, these give you everything needed to construct ℤ, ℝ, continuous functions, topological spaces, and virtually all objects in mainstream mathematics.
The nine axioms divide naturally into three groups. Some assert existence: the axiom of infinity guarantees an infinite set, pairing lets you form {a,b}, and power set gives you all subsets of a set. Some restrict or filter: separation prevents unrestricted comprehension, and regularity (also called the axiom of foundation) prevents sets from containing themselves, blocking certain pathological constructions. Some close sets under operations: replacement and union extend what you can build from sets you have.
The axiom of choice stands apart because it is independent of the others — it can be assumed or denied without creating a contradiction (assuming ZF itself is consistent, which we cannot prove from within ZF, per Gödel). Choice says that for any collection of nonempty sets, you can simultaneously pick one element from each — even for infinitely many sets with no defining rule for the selection. This seems obvious for finite collections but becomes subtle for uncountably many. Many important results in analysis and algebra require it, and some results that initially seem geometric or combinatorial turn out to secretly depend on it.