What are some practical uses of power series? Why do we care about power series? Why are they important?

Possibly the most basic reason why power series are useful is that they allow you to approximate any (possibly convoluted, but smooth) function $f(x)$ by its (truncated) power series $f(x) = \sum_n a_n x^n = \sum_{n \leq N} a_n x^n + O(x^{n+1})$. Power series are easy to manipulate and understand, general functions - not necessarily.
Say you want to understand the behaviour of $f(x) = \frac{x \sin x}{ 1 - \cos x}$ near $x = 0$. Just expand $\sin x = x - \frac{x^3}{6} + O(x^5)$, $\cos x = 1 - \frac{x^2}{2} +  \frac{x^4}{24} + O(x^6)$ and find that
$$f(x) = \frac{x^2 - \frac{x^4}{6} + O(x^6)}{\frac{x^2}{2} - \frac{x^4}{24} + O(x^6)}
= \frac{1- \frac{x^2}{6} + O(x^4)}{\frac{1}{2} - \frac{x^2}{24} + O(x^4)}
\\ = (1- \frac{x^2}{6} + O(x^4))(2 - \frac{x^2}{6} + O(x^4))
= 2 - \frac{x^2}{2} + O(x^4)
$$
So, with little computational effort, the complicated function that we start with turns out to behave "just like" the simple polynomial $2 - \frac{x^2}{2}$. In particular, it has limit at $0$ equal to $\lim_{x\to 0} f(x) = 2$, and is concave. If you wanted to say how much is $\int_0^t f(x) dx$, you could just integrate  $2 - \frac{x^2}{2}$, and the error would not be larger than $O(t^5)$. If you wanted to compare this to, say, $g(x) = 2 \cos x = 2 - x^2 + \dots$, then just looking at the coefficients you see that $f$ is greater, for $x$ sufficiently small. 
Of course, the same can be redone for higher degrees of accuracy if one needs smaller errors.
