definition of Axiom of Choice 
The Cartesian product of any nonempty collection of nonempty sets is
nonempty. In other words, if $I$ is any nonempty (indexing) set and
$A_i$ is a nonempty set for all $i\in I$, then there exists a choice
function from $I$ to $\cup_{i\in I}A_i$.

This is a text from Dummit and Foote. Can someone please explain what these sentences mean? I find the text very confusing. Is this a theorem or axiom?
Further, later in the text there is a theorem that says

Assuming the usual axioms of set theory, the following are equivalent:
(1) Zorn's Lemma (2) the Axiom of Choice (3) the Well Ordering
Principle

What are axioms of set theory and how are they related?
 A: Consider any set $I$. Suppose that for each $i \in I$, we specify some set $A_i$. We say that $A$ is an $I$-indexed family of sets.
We define the set $\prod\limits_{i \in I} A_i$ as $\{f \mid f : A \to \bigcup\limits_{i \in I} A_i$ and $\forall i \in I$, $f(i) \in A_i\}$.
In other words, a function $f \in \prod\limits_{i \in I} A_i$ is a function that takes as input some $i \in I$ and outputs an element of $a_i$.
This set $\prod\limits_{i \in I} A_i$ is the "cartesian product of the family $A$".
This generalises the notion of a binary Cartesian product $B \times C$. For we can define $A_0 = B$ and $A_1 = C$. Then $B \times C$ is in natural bijection with $\prod\limits_{i \in \{0, 1\}} A_i$.
The axiom of choice states that for all $I$, for every $I$-indexed family of sets $A$, if for all $i \in I$, $A_i$ has at least one element, then $\prod\limits_{i \in I} A_i$ also has at least one element.
For some reason, your text also imposes the requirement that $I$ not be empty. This requirement is just a distraction and is not necessary.
The "usual axioms of set theory" are probably the axioms of ZF.
A reasonable read for showing the equivalence of Zorn's lemma/axiom of choice/well-ordering theorem is here.
