The most elementary example is that the number of points on projective space $\mathbb P^n$ is always $(q^{n+1}-1)/(q-1)$. By the Lefschetz fixed points formula from topology applied to arithmetics, there's a following generalization for arbitrary variety $X$ over $F_q$.
There exist some algebraic numbers $\alpha_i$ with $|\alpha_i| = q^{n_i/2}$ for some $n_i$ such that $$\\# X(F_{q^n}) = \sum_i \pm \alpha^n_i.$$
Numbers $\alpha_i$ have specific properties and "come from geometry": they are eigenvalues of some operators on $H_{\text{et}}(X)$, thus there are as many of them as the dimension of this group; and these groups can often be directly compared to the case of $\mathbb C$.