What is something with 32 sides called?

Assuming you are talking about a polygon (and not a polyhedron, where "sides" would be faces), the thing is properly called either a "triacontakaidigon" or a "dotriacontagon" (your choice).

For more information about how to name polygons, see www.numericana.com/answer/culture.htm#polygons.

A 32-faced polyhedron may also be called an icosidodecahedron. The corresponding polygon is an icosidodecagon.

These particular words combine the greek words for twenty and twelve, to make 32.

Any others?

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BTW, DrG, enjoyed seeing your picture at the site!

Triacontakaidigon or Dotriacontagon. They are Polygons with 32 sides.

http://home.att.net/~numericana/answer/culture.htm
has a list of all names for different objects.

In my book, the term "icosidodecahedron" should only be reserved for the Archimedean polyhedron shown at right.  In particular, the term does NOT apply to another well-known Archimedean solid, the so-called "truncated dodecahedron", which also has 32 faces (20 triangular faces and 12 decagonal ones).

Only Eric W. Weisstein seems to be using the term "icosidodecagon" for a 32-sided polygon.  I feel it's a mistake (not consistent with standard Greek numerical prefixes).  If you know better, please let me know.  Thanks.

and I believe that Weisstein would agree with you: an icosidodecahedron is a 32-faced polyhedron that is specifically defined. Yours is the more appropriate general terminology for the polyhedron.

After all of these years, I must confess to having learned something on this post! I'm glad yours was the first answer, and hope that AlisonWonder used it.