What does it mean for two random variables to have bivariate normal distribution? The following is Sheldon Ross's definition:

We say that the random variables $X,Y$ have a bivariate normal
  distribution if, for some constants
   $\mu_x,\mu_y,\sigma_x>0,\sigma_y>0, -1&lt\rho &lt 1$, their joint density
  function is given, for all $-\infty &lt x,y &lt \infty$, by
  $$f(x,y)=\frac{\exp\left(-\frac1{2(1-\rho^2)}\left(\left(\frac{x-\mu_x}{\sigma_x}\right)^2+\left(\frac{y-\mu_y}{\sigma_y}\right)^2-2\rho\frac{(x-\mu_x)(y-\mu_y)}{\sigma_x\sigma_y}\right)\right)}{2\pi\sigma_x\sigma_y\sqrt{1-\rho^2}}$$

Is there a combinatorial/intuitive meaning of this definition?

There are several equivalent definitions of a random vector being multivariate normal. Every characterization  of a multivariate normal distribution is of course important. However sometimes it happens that one of the characterizations has an greater appeal from an intuitive standpoint. 
One such characterization is the following:
One can show that a random vector $(X,Y)$ is bivariate normal iff $(a_1,a_2)(X,Y)$  is normal for every vector $(a_1,a_2)$.
This result generalizes to higher dimensions, i.e. for random vectors of dimension $n$.
The nice thing about this definition of a multivariate normal variable is that many results become almost trivial to prove. For example it follows immediately that if X is multivariate normal, any marginal distribution of X is (multivariate) normal. 
