What does countable union mean?

It just means that every open set can be written in the form $$\bigcup_{i=1}^n(a_i, b_i)\qquad\text{or}\qquad\bigcup_{i=1}^{\infty}(a_i, b_i).$$ That is, every open set can be written as a union of either finitely many open intervals, or countably many open intervals.

It is a set of the form $\cup_{I \in S} I$ where $S$ is a countable set whose elements are open intervals.

We usually write $\cup_{k \in \mathbf{N}} I_k$, where $I_k$ is a sequence of intervals.

The formulations "union of a countable sequence of sets" and "union of a countable set of sets" are equivalent provided we have the axiom of choice.

In addition to the other answers, here is an example of an uncountable union:
Say that $A_x=(0,x)$ for every $x\in \mathbb R^+$. $$ \bigcup_{x\in \mathbb R^+} A_x $$ is an uncountable union.

By the Axiom of Union (in ZFC), if $A$ is a set, then there is a set $\bigcup A$ that is characterized by $x\in \bigcup A\iff\exists z\in A\colon x\in z$. We say "$X$ is a finite/countable union of foobar sets" if there exists a finite/countable set $A$ such that all elements of $A$ are foobar sets and $X=\bigcup A$.