"Spreading out" a smooth, connected $\mathbb{C}$-scheme of finite type.

For the general case, you need to know some of constructibility and openness results in EGA. But let me explain two "baby" cases which I hope will make clear what is going on.

${\color{red}{\text{Example 1:}}}$ We are given a monic polynomial $f(X)$ in one variable $X$, with coefficients in $\mathbb{C}$, and with all distinct roots. We want to find a finitely generated subring $R_0$ of $\mathbb{C}$ and a monic polynomial $f_0(X)$ with coefficieints in $R_0$ such that after the extension of scalars from $R_0$ back to $\mathbb{C}$, we get our original polynomial $f(X)$, and for every homomorphism from $R_0$ to a field, call it $k$, the polynomial over $k$ we get by applying the homomorphism to the coefficients of $f_0$ has all distinct roots (in an algebraic closure of $k$).

${\color{blue}{\text{First method:}}}$ Start with$$R_1 := \mathbb{Z}[\text{all the coefficients of }f].$$The discriminant $\Delta_1$ of $f_1$ is an element of $R_1$ which is nonzero in $\mathbb{C}$, so certainly nonzero in $R_1$. Take $R_0$ to $R_1[1/\Delta_1]$. What we have achieved is that the discriminant is now an invertible element of $R_0$.

${\color{blue}{\text{Second method:}}}$ Again start with $R_1$ as above. We know that over $\mathbb{C}$, $f$ and its derivative $f'$ generate the unit ideal in $\mathbb{C}[X]$. Explicitly, there exist complex polynomials $A$ and $B$ such that $Af + Bf' = 1$. Now adjoin to $R_1$ all the coefficients of both $A$ and $B$. Over this $R_0$, $f$ and $f'$ generate the unit ideal in $R_0[X]$.

${\color{red}{\text{Example 2:}}}$ We start with a nonsingular hypersurface in affine $n$-space, defined by one equation $f(X_1, \ldots, X_n) = 0$. The nonsingularity means that in $\mathbb{C}[X_1, \ldots, X_n]$, $f$ and its partial derivatives $df/dX_i$ generate the unit ideal. So there exist polynomials $A$, $B_1$, $B_2$, $\ldots$ , $B_n$ in $\mathbb{C}[X_1, \ldots, X_n]$ such that$$Af + \sum_i B_i {{df}\over{dX_i}} = 1.$$ Okay, start with$$R_1 := \mathbb{Z}[\text{coefficients of }f],$$then pass to$$R_0 := R_1[\text{all coefficients of }A\text{ and all the }B_i].$$Once you have a spreading out, say $X_1/R_1$ with structure map $\pi$ which is now smooth and whose $\mathbb{C}$-fiber is geometrically connected (which for a smooth scheme is the same as geometrically irreducible), you can use the fact for $n := \text{the relative dimension of }X_1/R_1$, and $\ell$ a prime invertible in $R_1$ (if there isn't one pass to $R_1[1/\ell]$ for your favorite $\ell$), $R^{2n}\pi_!(\mathbb{Z}/\ell\mathbb{Z})$ is constructible, so "locally constant" or lisse, on a dense open set of $\text{Spec}\,R_1$; Its rank at each point is the number of geometrically irreducible components of the fiber: as this rank is one at the point corresponding to $\mathbb{C}$ (a point lying over the generic point), this rank is one on an open dense set, so certainly over a set of the form $R_1[1/g]$ for some nonzero element $g$ in $R_1$.