I can remember twice that I have been really stunned by an equivalence of categories.
First, let R be a not necessarily commutative ring. The category of (left) R-modules is equivalent to the category of (left) modules over the ring of $n\times n$ matrices with entries in R. (This is the simplest example of a Morita equivalence.) I mean, the rings themselves look very different from each other.
The second is that the homotopy category of simplicial sets is equivalent to the homotopy category of CW-complexes. The former is defined purely combinatorially, and the latter is defined topologically: a CW complex is after all defined by gluing together open balls in $\mathbf R^n$ (albeit in a complicated way) and the equivalence relation on arrows given by homotopy is defined by taking products with the open interval [0,1]. It really is remarkable how the process of passing to homotopy can strip away all this topological structure.
