$$c\left(u_{1}, u_{2}\right)=\frac{1}{\sqrt{1-\rho_{12}^{2}}} \exp \left\{-\frac{\rho_{12}^{2}\left(x_{1}^{2}+x_{2}^{2}\right)-2 \rho_{12} x_{1} x_{2}}{2\left(1-\rho_{12}^{2}\right)}\right\}$$
is the copula density of the Gaussian copula, which is an elliptical copula, where $\rho_{12}$ is the parameter of the copula, $x_{1}=\Phi^{-1}\left(u_{1}\right), x_{2}=$ $\Phi^{-1}\left(u_{2}\right)$ and $\Phi^{-1}(\cdot)$ is the inverse of the standard univariate Gaussian distribution function.
$$
c\left(u_{1}, u_{2}\right)=\left(1+\delta_{12}\right)\left(u_{1} \cdot u_{2}\right)^{-1-\delta_{12}} \times\left(u_{1}^{-\delta_{12}}+u_{2}^{-\delta_{12}}-1\right)^{-1 / \delta_{12}-2}
$$
is the copula density of the Clayton copula, which is an Archimedean copula,
where $0<\delta_{12}<\infty$ is a parameter controlling the dependence. Perfect dependence is obtained when $\delta_{12} \rightarrow \infty,$ while $\delta_{12} \rightarrow 0$ implies independence.
Question
As you can see the elliptical copula density, despite being a function of $x_1$ and $x_2$, has the actual random variables $u_1$ and $u_2$ on the right hand-side, whereas the Archimedean copula densities do not have $x_1$ and $x_2$ on the right hand side, but instead what we would expect from the function: $u_1$ and $u_2$. This property seems to be a shared between all elliptical versus Archimedean copula densities. Why?
Source of formulas: Aas et al 2009 "Pair-copula constructions of multiple dependence"
Best Answer
It is not the case that one copula density has random variables as arguments and the other does not. A copula density is just a joint PDF where the underlying random variables have uniform marginal distributions.
In general, for a probability space $\Omega$ and random variables $Y_1: \Omega \to [0,1]$ and $Y_2: \Omega \to [0,1]$ the joint PDF $f$ is a function mapping the range $[0,1]\times[0,1]$ of $(Y_1,Y_2)$ into $\mathbb{R}$, that is $f:[0,1] \times [0,1] \to \mathbb{R}$, such that for real numbers $y_1,y_2 \in [0,1]$
$$F(y_1,y_2) =\mathbb{P}(\{\omega \in \Omega | Y_1(\omega) \leqslant y_1\} \cap \{\omega \in \Omega | Y_2(\omega) \leqslant y_2\}) = \int_0^{y_1} \int_0^{y_2} f(t_1,t_2)\, dt_1 \, dt_2$$
With regard to the copula density $c(u_1,u_2)$, the arguments $u_1$ and $u_2$ are always real numbers in $[0,1]$. This is a joint PDF for random variables $U_1$ and $U_2$ with uniform marginal distributions.
What may have been confusing is that the notation for the Gaussian copula does not show the explicit dependence on $u_1$ and $u_2$ on the RHS. The inverse of the normal CDF $\Phi^{-1}$ is a mapping from $[0,1]$ to $\mathbb{R}$ and the intermediate variables $x_1$ and $x_2$ are images $x_1 = \Phi^{-1}(u_1)$ and $x_2 = \Phi^{-1}(u_2)$.
The Gaussian copula density can be written more clearly as
$$c\left(u_{1}, u_{2}\right)=\frac{1}{\sqrt{1-\rho_{12}^{2}}} \exp \left\{-\frac{\rho_{12}^{2}\left([\Phi^{-1}(u_1)]^{2}+[\Phi^{-1}(u_2)]^{2}\right)-2 \rho_{12} \Phi^{-1}(u_1)\Phi^{-1}(u_2)}{2\left(1-\rho_{12}^{2}\right)}\right\}$$
A two-dimensional copula is a function $C:[0,1] \times [0,1] \to [0,1]$ with the following properties. For all $u_1,u_2 \in [0,1]$,
$$C(u_1,0) = 0, \,C(0,u_2) = 0,\\C(u_1,1) = u_1, \,C(1,u_2) = u_2,$$
and for all $0 \leqslant u_1 \leqslant u_1' \leqslant 1$ and $0 \leqslant u_2 \leqslant u_2' \leqslant 1$,
$$C(u_1',u_2') - C(u_1',u_2) - C(u_1,u_2') + C(u_1,u_2) \geqslant 0$$
Archimedean copulas are a just general class of such functions -- with no relationship to the normal distribution function $\Phi$.
In general, suppose we have random variables $X$ and $Y$ with joint CDF $F_{XY}$ and marginal CDFs $F_X$ and $F_Y$. we can obtain the copula associated with this joint distribution as the function $C: [0,1] \times [0,1] \to [0,1]$ given by the composition
$$\tag{*}C(u_1,u_2) = F_{XY}(F_X^{-1}(u_1),F_Y^{-1}(u_2))$$
As long as we start with a valid joint CDF, the function defined by (*) will always meet the requirements of a copula specified above.
A special case is the bivariate Gaussian copula. Letting $\Phi$ denote the univariate normal CDF and $\Phi_2$ denote the bivariate normal CDF, the bivariate Gaussian copula is given by
$$C(u_1,u_2) = \Phi_2(\Phi^{-1}(u_1), \Phi^{-1}(u_2))$$