Trying to understand an abstract definition of power series.. Can anyone explain to me what the $X_{n}$ and $X_{n}^{m}$ are really standing for?
Starting from "We write X for the power series..."

 A: Just like that of polynomials (not polynomial functions), the definition of formal power series is simply that they're sequences, except unlike polynomials they can be nonzero infinitely many times.
If you see a sequence $(a_n)$ as a power series, it is meant to represent an "infinite polynomial" that would look like this $\displaystyle\sum_{k=0}^\infty a_k X^k$.
The definitions of the product, and the sum of two power series follows from this "interpretation", and so do those of $X$ and $X^n$. Indeed, firstly, assume you want to describe the power series $X$ as a sequence $(a_n)$. Then looking at the infinite sum, you want $a_1=1$, and $a_n = 0$ for $n\neq 1$. Thus you put the $n$th term of $X$ (which is a sequence, so this $n$th term is $X_n$) to be $1$ if $n=1$, $0$ otherwise. Then you do the same for $X^m$, which gives you the formula $X^m_n = 1$ if $n=m$, $ 0$ otherwise (remembering that $X^m_n= (X^m)_n$ is the $n$th term of the sequence $X^m$. 
Then one can check that $X^m$ is indeed $X\times...\times X$, $m$ times with the definition of the product that was given. 
But first let me explain (I think I understood that you had trouble with this as well) why the product is defined as such. Consider $(p_n)$ and $(q_n)$ two sequences. We want to define their product in such a way that coincides with the interpretation that I gave earlier. 
We want $(p_n)(q_n)=(c_n)$ to coincide with $\left(\displaystyle\sum_0^\infty p_nX^n\right)\left(\displaystyle\sum_0^\infty q_nX^n\right) = \displaystyle\sum_0^\infty c_nX^n$.
Let's "calculate" (this is just to explain to you the definition, this is not really rigourous since there is no notion of convergence) the LHS: 
$\left(\displaystyle\sum_0^\infty p_nX^n\right)\left(\displaystyle\sum_0^\infty q_nX^n\right)= \displaystyle\sum_{(k,j)\in \mathbb{N}^2}p_k q_j X^{k+j}$, then we rearrange the terms according to the value of $k+j$ which gives $ \displaystyle\sum_{n=0}^\infty \displaystyle\sum_{(k,j)\in \mathbb{N}^2, k+j=n}p_k q_j X^n$. So this gives $c_n = \displaystyle\sum_{k+j=n}p_k q_j$, which can be rearranged to give $c_n =\displaystyle\sum_{k=0}^n p_k q_{n-k}$ .
Next (I'll leave that to you) you can prove, using this definition of product, by induction, that $X$ multiplied $m$ times with itself does give $X^m$ as defined earlier, which justifies the notation. At first, I recommend you keep in mind the fact that you're dealing with sequences, but I think that when you get more experienced you can "forget" that they're sequences, and really "think of them as power series". 
Let me know if there's anything I missed
