Let's start with the most elementary example: projective space $\mathbb P^n$. It's not hard to see that that the number of points on it is always $(q^{n+1}-1)/(q-1).$
Note that this is because $\mathbb P^n$ can be always decomposed into simpler pieces: $\mathbb A^n \cup \mathbb A^{n-1}\cup\dots\cup \mathbb A^0$. Interestingly, something similar applies to all $\mathbb F_q$-varieties. Specifically, the Lefschetz fixed points formula from topology applied to arithmetics gives the following statement for a variety $X/\mathbb F_q:$
There exist some algebraic numbers $\alpha_i$ with $|\alpha_i| = q^{n_i/2}$ for some $(n_i)$ such that the number of points $$\\# X(\mathbb F_{q^l}) = \sum_i (-1)^{n_i}\alpha^l_i\quad \text{for} l > 0 .$$
Numbers $\alpha_i$ in fact "come from geometry": they are eigenvalues of some operators acting on etale cohomology groups $H_{et}(X)$. In particular, the numbers $n_i$ can only occupy an interval between 0 and $\text{dim}\\, X$ and there are as many of them as the dimension of this group.
These groups can often be directly compared to the case of $\mathbb C$. To see how, consider the example of curves. Over $\mathbb C$ the cohomology have the form $\mathbb C \oplus \mathbb C^{2g} \oplus \mathbb C\ $ for some $g$ called genus; the same holds over $\mathbb F_q$:
- projective line $\mathbb P^1$ has genus 0, so it always has $n+1$ points
- elliptic curves $x^2 = y^3 + ay +b$ have genus 1, so they must have exactly $n + 1 + \alpha + \bar\alpha$ points for some $\alpha\in \mathbb C$ with $|\alpha| = \sqrt q.$ This is exactly the Hasse bound mentioned in another post.
These theorems laid foundation for studying varieties over $\mathbb F_q$ using the geometric intuition that comes from the complex case. You can read more at any decent introduction to arithmetic geometry or étale cohomology.