I don't know what's the difference between $\mathbb{C}[x]_{(x)}$ and $\mathbb{C}[x]$.
Isn't the localization is just equal to the original ring? Then why the first presentation is used?
I don't know what's the difference between $\mathbb{C}[x]_{(x)}$ and $\mathbb{C}[x]$.
Isn't the localization is just equal to the original ring? Then why the first presentation is used?
The ring $\Bbb C[x]_{(x)}$ is strictly larger. It is the localization of $\Bbb C[x]$ at the multiplicative subset $S=\Bbb C[x]\setminus (x).$
Since $\Bbb C[x]$ is a domain, $S$ has no zerodivisors, which implies that the natural homomorphism $\Bbb C[x]\to\Bbb C[x]_{(x)}$ is injective. In fact, since we obtain $\Bbb C[x]_{(x)}$ by formally inverting elements of $S,$ we can view both rings as subrings of $\Bbb C(x),$ the fraction field of $\Bbb C[x].$ Then $\Bbb C[x]_{(x)}$ consists of those $\dfrac{f(x)}{g(x)}$ such that $g(0)\neq 0,$ meaning the meromorphic function $\dfrac{f(x)}{g(x)}$ is locally regular at $x=0$ (which is exactly the closed point of $\operatorname{Spec}(\Bbb C[x])$ defined by the ideal $(x)$).
Since $(x)$ is prime, $\Bbb C[x]_{(x)}$ is a local ring. $\Bbb C[x]$ is of course not a local ring.