The algebraic fundamental group is the fundamental group of a scheme, as defined by Grothendieck in SGA1. It is essentially the fundamental group as seen by étale homotopy, the étale fundamental group.
For fields this is essentially the Galois group. For smooth varieties it is the Galois group of the maximal unramified extension of the function field of the variety (e.g. MK 09, p. 3).
In arithmetic geometry one also speaks of the arithmetic fundamental group.
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.
For more on this area, see at étale homotopy.
(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.)
or in a lengthier form:
An earlier version is to be found here.
A paper on a closely related subject is
See also
Minhyong Kim, Fundamental groups and Diophantine geometry, 2008 (pdf)
Minhyong Kim, Galois theory and Diophantine geometry, 2009 (pdf)