This result is apparently well known and used by many people.
I am, however, quite frustrated that I cannot seem to find a proof that I can understand.
For plane algebraic curves, this is not too hard.
For an irreducible polynomial $F(x,y) \in \mathbb{C}[x,y]$ nonconstant in both $x$ and $y$
with $F(0,0) = 0$,
one can formally expand $y$ as a Puiseux series in $x$
$$
y(x) = \sum_{i=0}^\infty y_i x^{\alpha_i}
$$
The Puiseux theorem actually states that such a series converges (in some sense) near $x=0$.
Alternatively, we can construct Riemann surface over the point $(0,0)\in \mathbb{C}^2$ of the algebraic function $y(x)$ which has the normal representation as holomorphic element
$$
\begin{eqnarray}
x &=& t^m \cr
y &=& \sum_{i=1}^\infty y_i t^i
\end{eqnarray}
$$
With some abuse of notation, I think we can even say the two are the same.
I can find many sources, among which I like Walker's representation the best.
Now how about space algebraic curves (of one complex dimension) in $\mathbb{C}^n$?
For a polynomial system
$F(x_0,\ldots,x_n) = (f_1(x_0,\ldots,x_n),\ldots,f_n(x_0,\ldots,x_n))$
where each $f_i \in \mathbb{C}[x_0,\ldots,x_n]$
with $F(0,\ldots,0) = (0,\ldots,0)$,
and lies only on a one (complex) dimensional irreducible component of $V(F)$,
if we fix a place of the this algebraic curve centred at $(0,\ldots,0)$,
we should be able to find a parametrization
$$
\begin{eqnarray}
x_0 &=& t^m \cr
x_k &=& \sum_{i=1}^\infty c_{k,i} t^i
\end{eqnarray}
$$
with convergent power series.
Or equivalently, we could express $x_1,\ldots,x_n$ as Puiseux series in $x_0$
that converge in certain sense.
I could only find "proofs" that reference Hironaka's resolution of singularity,
which I don't think I can understand any time soon.
I'm hoping to find a proof using only complex geometry or basic complex algebraic geometry.
In particular, I was thinking maybe I can repeatedly apply
Weierstrass preparation theorem together with Puiseux theorem,
however, I'm not quite sure how to continue after the first step.
Best Answer
If $(C,0)$ is a germ of complex curve in $\mathbb{C}^n$, then you can find coordinates $(z_1, z')$ and a polydisc $V=V_1\times V'$ centered in $0$ such that the canonical projection $V\ni (z_1,z')\mapsto \pi(z_1,z')=z_1\in V_1$ is a ramified covering when restricted to $C$ with $p$ sheets. Let $S$ be the ramification locus, then $S$ is an analytic set in $V_1$, that is, a discrete set of points. Therefore, upon taking a smaller $V_1$, we can think that $S=\{0\}$ or it is empty. Let us suppose $S$ is not empty, otherwise the projection is a local biholomorphism with the unit disc and this gives the thesis with $z_1=t$, $z_k=f_k(t)$, $f_k\in\mathcal{O}(D )$. Now, $C^*=C\cap \pi^{-1}(V_1\setminus S)$ is a $p-$sheeted covering of $V_1\setminus S=D\setminus\{0\}$, therefore it is isomorphic to the standard $p-$sheeted covering: $$D\setminus\{0\}\ni t\mapsto t^p\in D\setminus\{0\}.$$ This means that there exists a map $g:D\setminus\{0\}\to C^*$ such that $\pi\circ g (t)=t^p$; this map extends clearly to a holomorphic bijection by setting $g(0)=0$ (it is continuous in $0$ and holomorphic outside). Therefore, we have $$z_1=t^p$$ $$z_k=g_k(t)=\sum_{j=0}^\infty a_{kj}t^j.$$
The existence of such a system of coordinates is a standard result in local complex geometry, sometimes called local parametrization theorem.