In a recent blog post Terry Tao mentions in passing that:

"Class groups...are arithmetic analogues of the (abelianised) fundamental groups in topology, with Galois groups serving as the analogue of the full fundamental group."

Can anyone explain to me exactly in what sense are Galois and fundamental groups analogous?

In a recent blog post Terry Tao mentions in passing that:

"Class groups...are arithmetic analogues of the (abelianised) fundamental groups in topology, with Galois groups serving as the analogue of the full fundamental group."

Can anyone explain to me exactly in what sense are Galois and fundamental groups analogous?