Let $S$ be a connected scheme. Recall that a finite étale cover of $S$ is a finite flat surjection $X\to S$ such that each fibre at a point $s \in S$ is the spectrum of a finite étale algebra over the local ring at $s$. Fix a geometric point$\overline{s} : Spec(\Omega) \to \Omega$.

For a finite étale cover, $X\to S$, we consider the geometric fibre, $X\times_S Spec (\Omega)$, over $\overline{s}$, and denote by $Fib_\overline{s} (X)$ its underlying set. This gives a set-valued functor on the category of finite étale covers of $X$.

The algebraic fundamental group, $\pi_1(S, \overline{s})$ is defined to be the automorphism group of this functor.

(This entry is a stub and needs more work, including the linked entries that do not yet exist! Also explanation of $\Omega$. It is adapted from the first reference below.)