I know taylor series says that:

Let $\displaystyle f$ be a function with derivatives of all orders throughout some interval containing

$\displaystyle a$ as an interior point. Then the Taylor series generated by $\displaystyle f$ at $\displaystyle x = a$ is:

$\displaystyle \begin{align*}\sum_{k=0}^{\infty} \frac{f^{(k)}(a)}{k!}(x-a)^k = f(a) + f'(a)(x - a) +& \frac{f''(a)}{2!}(x-a)^2 + \cdots +\\& + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n + \cdots.\end{align*}$

So what do we mean when we say at $\displaystyle x = a$ for the above definition? And what is meant when we use the above definition for$\displaystyle x=0$?

Is it possible to kindly clarify these two (I'm not sure about the difference between two)?