What does representing a function as a power series actually mean ?

what it means is that any function that is reasonably well behaved in a certain interval can be approximated closely by an infinite sum of powers...meaning, let's take something as non algebraic as exp(x)...

this can be written as

exp(x)=1+x+x^2/2!+x^3/3!+x^4/4!+...

the more terms you add, the closer and closer the sum of those terms will come to equaling exp(x); if you are dealing with very small values of x, for instance, you are completely OK to evaluate exp(x) as 1+x since all higher powers of x will be too small to be meaningful

other well known power series are:

sinx = x-x^3/3!+x^5/5!-x^7/7!+...

cos x=1-x^2/2!+x^4/4!-x^6/6!+...

the concept of radius of convergence applies to describe the set of x for which these power series (so called because each term is a different power of x) are valid..for the power series above, the series are valid for all values of x...but for the power series

1/(1-x)=x+x^2+x^3+x^4+...
the series converges for values of abs x less than 1

As far as I know, no.

We like power series because it's just the sum of a bunch of things that look like x^n.  We know how x^n acts for any natural n, so all we have to worry about is how it converges.

Additionally, we can use it to approximate the values of various particularly difficult-to-evaluate functions.