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?

I don't have a combinatorial meaning, but you can think of it as follows.
$(X,Y)$ is the result of applying an affine transformation to a pair
$(W,Z)$ of independent standard normal random variables.  Many such
transformations exist, and one in particular is
$$\begin{align*}
X &= \mu_x + \sigma_x W\\
Y &= \mu_y + \rho \sigma_y W + \sqrt{1-\rho^2} \sigma_y Z
\end{align*}$$
See for example 
this set of slides.  The contours of the joint density (points at
equal height above the $x$-$y$ plane) are ellipses centered at $(\mu_x,\mu_y)$.
