Does every algebraic variety admit a local parametrization at every non-singular point? I am reading a text in which the first sentence of the proof of a theorem is:

Let $X(t)=X(t_{1},\ldots,t_{n})$ be a local parametrization of the algebraic variety $X$...

I guess that every algebraic variety admits certain kind of parametrization at every non-singular point, but I have not been able to find a book where this is proven. I would appreciate if you could tell me a book in which this is explained, and also if there is some relation between the tangent space at a point and its local parametrizations.
By the way, I know this is true in the case of plane algebraic curves, in which every point admits a local parametrization via Puiseux series.
 A: If you are working complex analytically, then the answer is yes because a smooth algebraic variety is a complex manifold, so if you have a smooth point, there exists some analytic disc around that point that is a manifold chart and therefore has coordinates in some open subset of $\mathbb{C}^n$.  
If you are working over a field (say algebraically closed of characteristic zero for simplicity) but not complex analytically, then you can still make this precise due to the following well known fact.  
Suppose $(R,\mathfrak{m})$ is a regular local ring with residue field $k = R/\mathfrak{m}$ so that $k$ has the same characteristic as $R$. Then the completion $\widehat{R}$ is isomorphic to 
$$
k[[t_1, \ldots, t_n]]
$$
where $n = \dim R$ is the Krull dimension. 
So for an algebraic variety $X$ over $k$ at a nonsingular point $x \in X$ we can take the local ring $(\mathcal{O}_{X,x},\mathfrak{m}_x)$. Then $x$ is a nonsingular point if and only if $\mathcal{O}_{X,x}$ is a regular local ring if and only if $\widehat{\mathcal{O}}_{X,x} \cong k[[t_1, \ldots, t_n]]$ where $n = \dim X$. These $t_1, \ldots, t_n$ form your local parameters at a point. 
The fact that $k[[t_1, \ldots, t_n]]$ is the completion of a polynomial ring in the variables $t_i$ reflects that fact that smooth points look locally analytically like $\mathbb{A}^n$ (similar to how in the complex analytic case smooth points have an actual neighborhood isomorphic to an open subset of $\mathbb{C}^n$). 
These local parameters $t_i$ at $x$ are related to the tangent space by the fact that $T_xX = (\mathfrak{m}_x/\mathfrak{m}_x^2)^\vee$. The $t_i$ in the completion are exactly a choice of basis for $\mathfrak{m}_x/\mathfrak{m}_x^2$ as a $k$-vector space and so they are dual functions to the tangent space just as in the case of a manifold where your local parameters are dual to the tangent directions. 
