AC Adaptor broke when powering PC case fan
The adapter has an AC output.  Your fan expects DC.

I expect the AC adapter was "seeing" a short circuit for half of each AC wave and overheated.

Such adapters have a fusible link on the primary side (where it plugs into the wall.)

If the transformer gets too hot, the link opens and the transformer is dead - permanently.  This is on purpose.  It prevents a shorted transformer from burning down your house.



As Russell McMahon mentioned in the comments, the ~ is a give away in the output specification.

All together, there are a few things that point to the output being AC:


The ~ in the output voltage specification.  DC would have a symbol that looks more like a couple of equal  signs(==).
It specifies a primary and secondary voltage.  Secondary is a term that applies to transformers, so the output will be AC.  Power supplies with DC output usually say "Output" rather than secondary 
The output power rating is in VA rather than watts.  VA is used in reference to AC power rather than DC.