The easiest example I can think of is the natural incidence correspondence between $\mathbb{P}^3$ and the parameter space of cubic surfaces. This can be used to show that every cubic surface contains a line; from this [it follows easily][1] that every smooth cubic surface contains exactly $27$ lines.

Another example is the moduli space of stable maps constructed by Kontsevich; this parametrizes certain maps from curves to (to stick with a simple case) $\mathbb{P}^2$. It can be used to answer the following question: given $3d-1$ points in $\mathbb{P}^2$ in general position, compute the number $N_d$ of *rational* curves of degree $d$ passing through these points. It turns out that the values $N_d$ satisfy a certain recursive relation which allows you to compute all these numbers starting from the obvious $N_1 = 1$ (through $2$ points passes exactly one line). You can find the formula at the entry [Kontsevich's formula][2] on [Rigorous trivialities](https://rigtriv.wordpress.com); it yields for instance $N_2 = 1$ and $N_3 = 12$.

Yet another example, again more elementary is the following. The Grassmannian $G = \operatorname{Gr}(1, \mathbb{P}^3)$ parametrizes lines in $\mathbb{P}^3$. The computation of the cohomology of $G$ allows you to compute the number of lines which are incident to $4$ fixed lines in general position (it turns out this number is $2$).


  [1]: https://mathoverflow.net/questions/20112/interesting-results-in-algebraic-geometry-accessible-to-3rd-year-undergraduates/20261#20261
  [2]: http://rigtriv.wordpress.com/2008/12/23/kontsevichs-formula/