The Taylor series of a real/complex-valued function $f(x)$ that is infinitely differentiable at real/complex value $a$ is as follows:
$$f(x) = \sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n$$
I know we use this in a lot of places but I have no idea why they're essential. If we are capable of taking derivatives of a function at a point, and we are capable of evaluating the function at a point, why do we need a summation or power series that only gives an approximation?
On
The usual situation:
You "know" $f$, $f'$, $f''$, $\dots$
But you don't know how to calculate $f(x)$ for any point $x$.
But you know how to calculate $f(a)$ for some concrete point(s) $a$.
You can write the Taylor series at $a$.
You can approximate $f(x)$ at least for $x$ near $a$.
Example: $f(x) = e^x$. $$\forall n\in\Bbb N: f^{(n)} = f,$$ $$\forall n\in\Bbb N: f^{(n)}(0) = f(0) = e^0 = 1,$$ $$f(x) = \sum_{n=0}^\infty\frac{x^n}{n!}.$$
From a theoretical standpoint, some times a Taylor series is all you have for a function. For instance, many differential equations can be solved by induction on the degree of the Taylor series.
As for practical uses, when you ask a computer to evaluate $\sin(4.3)$, say, then evaluating the Taylor series of the sine function with $x = 4.3$ up to some predetermined degree is what the computer actually does, because multiplication and addition is very easy for them (most modern processors have built-in special-purpose multiplying and adding circuits that do this really quickly; this is what FLOPS measures).