I don't understand this specific part of the textbook about countable union in the probability theory book 
Why is it called countable union? Is it because when all sets are in union, it all becomes one big set?
 A: 
Is it because when all sets are in a union, it all becomes one big set?

This isn't quite the reason.
Infinite sets come in different sizes.  The smallest size is the size of the natural numbers, or, as you said, any set that can be put in a bijective relationship with the natural numbers.  Such sets are called “countable” or “countably infinite”, because you can enumerate their elements in order, the first one, the second one, the third one, and so on.  (That's what the bijection with the natural numbers is doing.)
But some infinite sets are larger, and there is no bijection with the natural numbers.  Such sets are called “uncountable”.  An example is the set of real numbers.  You simply can't make a list of all real numbers; any such list leaves out most of the numbers.  (If this seems bizarre, it's because it is. It was discovered 150 years ago and we are still trying to make sense of it.)
Now suppose you have a family of sets, $\mathcal F$.  (Family means exactly the same as “set”, but we sometimes call a set of sets a “family” because it sounds less confusing.)  Each element of the family $\mathcal F$ is a set, maybe a set of points or numbers or something. $\mathcal F$ could be a finite family, in which case we might write something like $\mathcal F = \{ S_1, S_2, S_3 \}$ (if there are three sets in the family), or an infinite family.
However big $\mathcal F$ is, we can talk about the union of all the sets in $\mathcal F$.   Some element $x$ is in the union if it is an element of some set in $\mathcal F$.
If $\mathcal F$ is finite, we have what's sometimes called a "finite union”; a union of a finite family of sets.  Supposing just for example that the finite family consists of the three sets $S_1, S_2, S_3$, we might write the union as $$S_1 \cup S_2 \cup S_3$$ or $$\bigcup_{i=1}^3 S_i.$$
If $\mathcal F$ is infinite we can do something similar.  Say that $$\mathcal F=\{S_1, S_2, \ldots \}.$$ Then we can write the union as $$S_1 \cup S_2 \cup\ldots$$ or $$\bigcup_{i=1}^\infty S_i.$$
Ah ah, not quite! Those notations only make sense if $\mathcal F$ is a countable family of sets.  If it isn't, if it is an even bigger infinite family of sets, then it's too big for us to make a list $S_1, S_2, S_3,\ldots$.  That's what “uncountable” means.  In that case, those notations don't make sense, and we have to find another way to talk about the union.
The reason this is important in probability theory is that uncountable sets behave badly in many ways. (This is why we are still trying to figure them out.)  One way in which they're bad is this: for probability theory we are measuring how likely certain events are.  The measures are numbers called probabilities.  We want to say that if event $A$ has a probability of $a$ and event $B$ has a probability of $b$, then we can make a bigger event that consists of the union of $A$  and $B$, and it will have probability no more than $a+b$.
This works for finite unions, and it works for countable unions.  But it doesn't work for larger unions.  If you have an uncountable family of events that each have probability $0$, their union might have a probability of $1$.  This doesn't happen with countable unions.
So your textbook wants to stick with finite and countable unions because we don't know how to make probability theory work with bigger unions.
A: The union $\bigcup_{j\,=\,1}^\infty C_j$ is called a countable union because the set of indices $j\in\{1,2,3,\ldots\}$ is countably infinite rather than uncountably infinite.
A: A set $\mathcal{M}$ is called countable, if there is a surjective map $\psi:\mathbb{N}\to \mathcal{M}$. Consider now a family of sets $\{A_i: i\in I\}$ with some index set $I$. If $I$ is countable, we say the union $\bigcup_{i\in I}A_i=\{x: \exists i \in I \text{ s.t. } x\in A_i\}$ is a countable union. For example, if $A_i=(i,i+1]$ for $i\in\mathbb{N}$ we can write $\mathbb{R}_{>0}$ as countable union $\bigcup_{i\in \mathbb{N}}A_i$.
The definition of countable union in the textbook is what I would call "countably infinite" (because I would consider finite sets also as countable). Probability theory is about measuring probabilities, so we need to find a way to define measure. This sounds easier than it is, there are some paradoxes which arise if we do not define measures properly (https://en.wikipedia.org/wiki/Vitali_set or https://en.wikipedia.org/wiki/Banach–Tarski_paradox).
Todays definition of probability measures relies on Kolmogorov's axioms (https://en.wikipedia.org/wiki/Probability_axioms#Axioms) which relies on an underlying measure space. So, if the $A_i$ are some events with a certain probability, we want a countable union to be again an event for which a probability can be measured.
