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?

You should think of coverings of manifolds as analogous to field extensions.  Once you accept this, then the fundamental group and absolute Galois group play the same role; coverings correspond to subgroups of the former and field extensions to subgroups of the latter (though for the absolute Galois group you have to consider its topology).

This can be made precise in algebraic geometry: if you have a covering map of projective algebraic varieties, then the function field of the target embeds into the function field of the domain by pullback, and this is a finite degree unramified field extension. 

You can think of lifting paths downstairs as being a bit like algebraic number theory: each closed path downstairs has an inverse image that's a union of paths.  If the covering is Galois, then each component will cover the original with the same degree, but otherwise maybe not.  You can think of the conjugacy class of the path as the "Frobenius" whose orbit type on the set of preimages of a point determines the "splitting into primes."

There's even a version of the theory of L-functions given by considering the spectrum of the Laplacian for a metric on the varieties.
