copularandom-generation
Suppose that I have a 2-dim copula function C(x_1,x_2).
How can I generate bivariate numbers from this copula?
For specific types of copulas, I can use 'rCopula' function of 'copula' package in R.
But I have no idea what to do if I have an arbitrary copula function.
A typical approach (see e.g. Nelsen 2006, p. 41) is to sample two independent uniform distributed random vectors $u$ and $y$ of the desired sample length. The conditional copula $C_u$ (conditioned on $u$) is given through the partial derivative: $$ C_u(v) = \frac{\partial}{\partial u} C(u,v) $$ Hence, one needs to solve $C_u(v)=y$ for $v$ to get the desired pair $(u,v)$. For a "custom made" copula, one has to calculate its partial derivative and its quasi-inverse. In case the copula is not completely "custom made" it might already be covered in other statistical software. One might for instance take a look into the R packages copula and VineCopula offering a rich set of families (speaking from my R experience, there are more in R and of course in other languages).
Thanks for your answers. Indeed, when I plot the different Copulas, it seems (based on a visual comparison) that the dependence struture is depicted in the copula.
Then my next step will be to compare the different copula families in terms of looking at their goodness of fit.
@Ben: Ok, so if the distribution of one of my marginals is the generalized extreme value distribution, my u or v will be passed to the "x" of this function, since the other parameters of this distribution are fitted before. 
Best Answer
For a copula that corresponds to a known multivariate distribution, you can simulate from that distribution and then make the margins uniform (e.g. Gaussian copula, t-copula).
More generally if you can work out the conditional (either $C(u|v)$ or $c(u|v)$), you can simulate from a uniform for $V$ and then from the conditional, perhaps via inverse-cdf (if you know $C(u|v)$) or perhaps via say accept-reject (maybe an adaptive accept-reject, some version of ziggurat, etc, if you know $c(u|v)$).
In the case of bivariate Archimedean copulas, following Nelsen (1999) or Embrechts et al., (2001), we have a mechanism for then generating from them as follows. Suppose $(U_1,U_2)$ has a two-dimensional Archimedean copula with generator $\phi$. Then:
Simulate two independent $U(0,1)$ random variables, $v_1$ and $v_2$
Set $t=K_C^{-1}(v_2)\,$, where $K_C(t)=t-\phi(t)/\phi'(t)$
The desired simulated values are $u_1=\phi^{-1}(v_1\,\phi(t))$ and $u_2=\phi^{-1}((1-v_1)\phi(t))$.
There are other methods; for example in some cases it might sometimes be practical to do some version of bivariate accept-reject, say, or via transformation to some convenient bivariate distribution on which accept-reject might be applied.