What is the role of the random catalogue in calculating the two-point correlation function? To study clustering of galaxies and clusters, one makes use of
the two-point correlation function. However, we can see that all 
the estimators of $\xi(r)$ use a random catalogue. 

What is the random catalogue used for?
The point of the two-point correlation function (pun not intended) is to describe how clustered the galaxies in the universe are. Astronomers want to know if they're all bunched up in tight bundles with huge voids, or maybe they're more or less uniformly distributed.
The two-point correlation function is a way of measuring the distribution of these galaxies compared to a random distribution. In other words, if your distribution of galaxies is not uniformly random (which it isn't for our universe - our universe has a distinct non-random structure of clusters and voids) then your correlation function will indicate the degree to which it isn't random. Let's take a look at the last equation listed in your link.

The most commonly-used estimator is from Landy &amp; Szalay (1993)
$$\xi(r) = \frac{1}{RR(r)}\Big[DD(r)\Big(\frac{n_R}{n_D}\Big)^2 - 2DR(r)\Big(\frac{n_R}{n_D}\Big)+RR(r)\Big]$$

The symbols in this equation are defined as follows:

$\xi(r)$ - The two-point correlation function.
$r$ - The separation between two galaxies.
$DD$,$DR$,$RR$ - The counds of pairs of galaxies as a function of separation in the data catalogue, between the data and random catalogue, and in the random catalogue, respectively.
$n_D$, $n_R$ - The mean number density of galaxies in the data and random catalogues, respectively.

If your data catalogue happens to be completely random, then $DD=DR=RR$ and you find that $\xi(r)=0$ (assuming also that $n_D=n_R$), showing your data catalogue is indeed indistinguishable from a random distribution. If, however, your data catalogue is not defined with completely random galaxies, you'll find $\xi(r)\ne0$, and the degree to which it is non-zero indicates how non-random, or organized your system is.