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 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 here; it yields for instance $N_2 = 5$ 1$and$N_3 = 12$. Yet another example, again more elementary is the following. The Grassmannian$G = \mathop{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$2$). 2 added 1055 characters in body 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 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 here; it yields for instance$N_2 = 5$and$N_3 = 12$. Yet another example, again more elementary is the following. The Grassmannian$G = \mathop{Gr}(1, \mathbb{P}^3)$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 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 that every smooth cubic surface contains exactly$27\$ lines.