For concreteness, let me provide an explicit description of the ring in Roman's answer above for $G=GL(n)$. There are several versions of universal centralizers in their paper, and I will describe two of them.
1. $$X=\{(x,y)\in \mathfrak{gl}(n)\times \mathfrak{gl}(n): x\ \text{regular},\ [x,y]=0\}/GL(n)$$
We can think of $x$ and $y$ as two commuting matrices where $x$ is regular. We let $x_i$ be the eigenvalues of $x$, and consider the following polynomials:
$a(x)=\prod (x-x_i)$ is the minimal polynomial of $x$, it is a monic polynomial of degree $n$ (since $x$ is regular).
$b(x)=\sum_{i=0}^{n-1} b_i x^i$ is a (not necessarily monic) polynomial of degree at most $n-1$ such that $b(x)=y$. Such a polynomial exists since $x$ is regular and $[x,y]=0$.
The coefficients of $a(x)$ and $b(x)$ are the coordinates on $X$. There is an action of $S_n$ permuting $x_i$ and fixing $b_i$, and we can also write
$$
\mathbb{C}[X]=\mathbb{C}[x_1,\ldots,x_n,b_0,\ldots,b_{n-1}]^{S_n}
$$
We would like to compare this to the blowup construction. Let $$A=\mathbb{C}[x_1,\ldots,x_n,y_1,\ldots,y_n]^{sgn}\subset \mathbb{C}[x_1,\ldots,x_n,y_1,\ldots,y_n]$$ be the space of antisymmetric polynomials with respect to the diagonal action of $S_n$, and $\Delta=\prod_{i<j} (x_i-x_j)$ Then
$$
\mathbb{C}[X]\simeq \mathbb{C}\left[\frac{\alpha}{\Delta}: \alpha \in A\right]\subset \mathrm{Frac}\mathbb{C}[x_1,\ldots,x_n,y_1,\ldots,y_n].
$$
This is a localization of the graded ring $\bigoplus_{d} A^d$ where we assume that $A^0$ is the space of symmetric polynomials. The graded ring corresponds to the simultaneous blowup that Roman mentioned.
Proof: Define $y_i=b(x_i)$. First, we can write $b_i$ as rational functions of $x_i$ and $y_i$ as follows. We solve the system of equations $b_0+b_1x_i+\ldots+b_{n-1}x_i^{n-1}=y_i$ by Cramer's Rule and get
$$
b_i=\mathrm{Alt}\left(x_1^0\cdots \widehat{x_i^{i-1}}\cdots x_n^{n-1}\cdot y_i
\right)/\Delta.
$$
These formulas hold on the locus when $x_i$ are distinct. However, we know that $b_i$ uniquely extend to regular functions on $X$ and generate the ring of functions.
Thus we get a well defined map from $\mathbb{C}[X]$ to the localization of $\bigoplus_d A^d$.
To construct the inverse map, we need to express arbitrary ratio $\frac{\alpha}{\Delta}$ for $\alpha\in A$ as a symmetric polynomial of $x_i,b_i$. By denoting $y_i=b(x_i)$, we can write any polynomial in $x_i,y_i$ as a polynomial in $x_i,b_i$. The action of $S_n$ permutes $x_i$ and $y_i$ simultaneously and hence agrees with the action on $\mathbb{C}[x_1,\ldots,x_n,y_1,\ldots,y_n]$. Now $\alpha(x_1,\ldots,x_n,y_1,\ldots,y_n)$ can be written as a polynomial in $b_i$ with coefficients being antisymmetric polynomials in $x_i$, hence it is divisible by $\Delta$ and $\alpha/\Delta$ is as a polynomial in $b_i$ with coefficients being symmetric polynomials in $x_i$, so it is a well defined element of $\mathbb{C}[X]$.
This argument is essentially borrowed from a paper by Mark Haiman:
"t,q-Catalan numbers and the Hilbert scheme
Discrete Math. 193 (1998), 201-224."
In particular, $X$ can be identified with the open chart $U_{(n)}$ in the Hilbert scheme of points on the plane.
2. We can generalize the above arguments to the more complicated universal centralizer
$$Y=\{(x,y)\in \mathfrak{gl}(n)\times Gl(n): x\ \text{regular},\ [x,y]=0\}/GL(n)$$
The setup is as above but now $y$ is invertible. We define three polynomials:
- $a(x)=\prod (x-x_i)$ is the minimal polynomial of $x$
- $b(x)=\sum_{i=0}^{n-1} b_i x^i$ is a (not necessarily monic) polynomial of degree at most $n-1$ such that $b(x)=y$.
- $b^*(x)=\sum_{i=0}^{n-1} b^*_i x^i$ is a (not necessarily monic) polynomial of degree at most $n-1$ such that $b^*(x)=y^{-1}$.
We have an action of $S_n$ permuting $x_i$ and fixing $b_i,b^*_i$ and
$$
\mathbb{C}[Y]=\left(\frac{\mathbb{C}[x_1,\ldots,x_n,b_0,\ldots,b_{n-1},b^*_0,\ldots,b^*_{n-1}]}{(b(x_i)b^*(x_i)-1,\ i=1,\ldots,n)}\right)^{S_n}.
$$
Note that we can rephrase these conditions by saying that $b(x)$ does not vanish at roots of $a(x)$, or that the polynomials $a(x)$ and $b(x)$ are coprime (and then $b^*(x)$ is determined), but the above description is easier to work with. See also https://arxiv.org/abs/1503.04817, Section 5.2 for a similar discussion and a connection to the moduli space of monopoles.
The blow-up description above generalizes more or less verbatim. Indeed, now
$A\subset \mathbb{C}[x_1,\ldots,x_n,y_1^{\pm},\ldots,y_n^{\pm}]$ is the space of antisymmetric polynomial and $\mathbb{C}[Y]$ is the localization of the graded ring $\bigoplus_d A^d$ in $\Delta$. We can define
$$
y_i=b(x_i),\ z_i=b^*(x_i)
$$
and by the above equations $y_i\cdot z_i=1$ and $z_i=y_i^{-1}$. Similarly to the above, we can solve for $b_i$ in terms of $y_i$ and $x_i$, and for $b_i^{*}$ in terms of $y_i^{-1}$ and $x_i$:
$$
b^*_i=\mathrm{Alt}\left(x_1^0\cdots \widehat{x_i^{i-1}}\cdots x_n^{n-1}\cdot y_i^{-1}\right)/\Delta.
$$
The rest of the argument generalizes verbatim.
