"A Galois group is a fundamental group"?

Solution 1:

That comment refers to the étale fundamental group of a scheme, which is a more subtle notion than the usual fundamental group. As stated in the comments, a thorough introduction to this point of view can be found in Szamuely's Galois Groups and Fundamental Groups.

The basic idea is that one should think of the category of finite extensions of a field $K$ as being analogous to the category of finite coverings of a topological space; the Galois group and fundamental group, respectively, come from trying to understand these categories. This analogy is closest in the case that $K$ is a one-dimensional function field over $\mathbb{C}$; in that case, it turns out that $K$ is the field of meromorphic functions on a compact Riemann surface, and that studying finite extensions of $K$ is the same thing as studying (branched) covers of this Riemann surface.

Solution 2:

There is a nice survey article by Liang Xiao in here.