What Shafarevic calls a local parameter is often called a uniformizing parameter at $P$, and is also the same thing as a uniformizer of the local ring of $C$ at $P$.
The point is that if $P$ is a smooth point on a curve, then the local ring at $P$ (i.e. the ring of rational functions on $C$ which are regular at $P$) is a DVR, and hence its maximal ideal is principal; a generator of this ideal is called a uniformizer.
If $t$ is a uniformizer/local parameter/uniformizing parameter at $P$, and if
$u$ is any other rational function, then if we write $u = t^k v$ where $v(P) \neq 0$ (i.e. $v$ is a unit in the local ring), then $k$ is the order of vanishing of $u$ at $P$. In particular, $u$ vanishes to order one if and only if it is equal to $t$ times a unit in the local ring, if and only if it is also a generator of the maximal ideal of the local ring at $P$, if and only if it is also a uniformizer. Thus Shafarevic and Wikipedia are reconciled.
One is supposed to think of $t$ as being a "local coordinate at $P$." In the complex analytic picture you would choose a small disk around $P$, and consider the coordinate $z$ on this disk; this a local coordinate around the smooth point $P$. This analogy is very tight: indeed, it is not hard to show (when the ground field is the complex numbers) that a rational function $t$ is a local parameter at $P$ if and only $t(P) = 0$,
and if there is a small neighbourhood of $P$ (in the complex topology) which is mapped isomorphically to a disk around $0$ by $t$, i.e. if and only if $t$ restricts to a local coordinate on a neighbourhood of $P$.
Finally, this concept is ubiquitous. The fact that the local ring at a point on a smooth algebraic curve is a DVR is fundamental in the algebraic approach to the theory of algebraic curves; see e.g. section 6 of Chapter I of Hartshorne.
$\newcommand{\Reals}{\mathbf{R}}$Theorem: Let $S^{2} \subset \Reals^{3}$ be the unit sphere centered at the origin, and $n = (0, 0, 1)$ the north pole. The stereographic projection mapping $\Pi:S^{2} \setminus\{n\} \to \Reals^{2}$ is conformal.
Proof: Fix a point $p \neq n$ arbitrarily, and let $v$ and $w$ be arbitrary tangent vectors at $p$. The plane containing $n$ and $p$ and parallel to $v$ cuts the sphere in a circle $V$ tangent to $v$. Similarly, the plane containing $n$ and $p$ and parallel to $w$ cuts the sphere in a circle $W$ tangent to $w$.
The circles $V$ and $W$ form a digon with vertices $p$ and $n$. By symmetry, the angle $\theta$ they make at $p$ is equal to the angle they make at $n$. Let $v'$ and $w'$ be the tangent vectors at $n$ obtained from $v$ and $w$ by reflection across the plane of symmetry that exchanges $p$ and $n$.
Because the tangent plane to the sphere at the north pole is parallel to the $(x, y)$-plane, the image $\Pi_{*}v$ of $v$ under $\Pi$ is parallel to the vector obtained by translating $v'$ along the ray from $n$ through $p$ to the $(x, y)$-plane. Similarly, $\Pi_{*}w$ is parallel to the vector obtained by translating $w'$.
It follows at once that the angle between $\Pi_{*}v$ and $\Pi_{*}w$ is equal to the angle between $v'$ and $w'$, which is equal to the angle between $v$ and $w$.
Conformality of stereographic projection allows the holomorphic structure on the Riemann sphere to be visualized in three-dimensional Euclidean geometry, providing a basic link between complex analysis and differential geometry.
In coordinates, stereographic projection and its inverse are given by
$$
\Pi(x, y, z) = \frac{(x, y)}{1 - z},\qquad
\Pi^{-1}(u, v) = \frac{(2u, 2v, u^{2} + v^{2} - 1)}{u^{2} + v^{2} + 1}.
$$
It's possible to calculate the induced Riemannian metric on the plane:
$$
g = \frac{4(du^{2} + dv^{2})}{(u^{2} + v^{2} + 1)^{2}}.
$$
Conformality is encoded in $g$ being a scalar function times the Euclidean metric. While this analytic argument has its own elegance, the geometric argument above is essentially obvious.
Best Answer
Well, you'll really want to read them all at some point. To start with, take Griffiths-Harris for geometric insight and Huybrecths for company (his chapter 1.2 is amazing). Voisin is very good and at first covers the same ground as Huybrecths, but is more advanced (do read the introduction to Voisin's book early, it sets the scene quite well). Demailly's book is where all the details are, you'll want that one for proofs of the main theorems like Hodge decomposition, Kodaira vanishing etc. There's also a new book by Arapura that looks very user-friendly.
And now for some clearly false generalities: The books by Huybrechts, Voisin and Arapura have very algebraic points of view; they were written by people who are mainly algebraic geometers and (to simplify greatly) think in Spec of rings. By contrast, Demailly and Griffiths-Harris have more differential-geometric points of view and use metrics and positivity of curvature as their main tools. I'll take the opportunity to also recommend Zheng's wonderful "Complex differential geometry" for an alternative introduction to that point of view. You'll need to know how to use all of these tools (as do all those people, of course).
So, to sum things up: $$ \begin{array}{ccc} & \hbox{introduction} & \hbox{advanced} \\ \hbox{algebraic} & \hbox{Arapura, Huybrechts} & \hbox{Voisin}\\ \hbox{metric} & \hbox{Griffiths-Harris, Zheng} & \hbox{Demailly} \\ \end{array} $$