Axiom of choice in set theory

Just as the title stated, what is the main point of axiom of choice? I do not quite understand what is written in the axiom. The axiom that I know is:

Given any collection of non-empty sets, there exists a choice function such that $$f:I \rightarrow \bigcup_i{S_i}\quad f(i)\in S_i \quad\text{for all }\; i\in I.$$

The point of the Axiom of Choice is that oftentimes a mathematician finds him- or herself at a point where infinitely many specific objects must be gathered up all at once in order to continue a proof/construction.

In the statement as given, we have a family of nonempty sets $\{ A_i : i \in I \}$, and we want to pick a unique representative from each $A_i$, this is our function $f : I \to \bigcup_{i \in I} A_i$ ($f$ "picks" $f(i)$ to be the unique representative from $A_i$). The Axiom of Choice says that this is unproblematic, and we can always do this.

Of course, there are certain specific instances where one can do this without appealing to the Axiom of Choice:

• if you only have to make finitely many choices; or
• if these choices can be made in a uniform manner (e.g., if I have infinitely many nonempty sets of natural numbers, I can choose the least one from each set).

More often there is no way to uniformly pick these representatives, and without appealing to some extra-logical hypothesis we cannot make the choices as required.

(There are many statements known to be equivalent to the Axiom of Choice. The most common one you see in mathematics outside of logic/set theory is Zorn's Lemma.)

The axiom of choice states that when given a family of nonempty sets, we can choose an element from each set. Hence the name, the axiom of choice.

It is useful because it allows us to control the behavior of infinite objects. It ensures that we can prove the existence of things which otherwise may require an infinitely long statement, for example a Hamel basis for every vector space.

When writing a proof about an infinite collection of objects we will sometimes want to make some arbitrary choice amongst them and just "run with it". It might be possible to make that choice less arbitrary, for example if we talk about finite sets of real numbers you can always take the minimal element; or if you talk about continuous functions from a compact interval to $\mathbb R$, you can talk about the smallest point which attains the maximal value.

When you have a uniform way of making a choice, that is one "algorithm" which ensures that a choice is made, then you really have no need for the axiom of choice. The axiom of choice supplies this method when you cannot prove it exists otherwise.