What do you call that thing that cleans up the ice on ice-skating rinks?

Beezer13,

According to the Central Park Conservancy the ice-cleaning machine
used to clean the ice at skating rinks is called a Zamboni.

"Today, the Park?s paths are still cleared the old-fashioned way ?
with shovels wielded by hard-working Conservancy field staff (they do
sometimes use hand-held snow blowers and snow plows), but the skating
rinks are cleared by a Zamboni, a large ice-cleaning machine named
after its inventor, Frank Zamboni."

The Central Park Conservancy 
http://www.centralparknyc.org/17613/22456

HEre are some Zamboni images
http://images.google.com/images?svnum=50&hl=en&lr=&rls=GGLD%2CGGLD%3A2005-18%2CGGLD%3Aen&q=zamboni


Search terms:
ice cleaning machine 
Zamboni


I hope this helps!

Best regards,
Bobbie7

Very good.  Thank you for your speed in answering also.

Zamboni is a registered trademark.  The generic term is "ice resurfacer".

http://www.zamboni.com/welcome.html

Definitions of zamboni on the Web:

the machine used to resurface the ice between periods. 
www.eiha.co.uk/glossary.html

The ice re-surfacing machine.
espn.go.com/nhl/s/rules/glossary.html

This is the name of a particular type of ice resurfacing machine but
the name is loosely used to refer to any type of ice resurfacing
machine.
www.iwyha.com/ice_wolf_hockey_terms.html

For the non-hockey initiated, it's the machine, invented by Frank
Zamboni, that scrapes and smoothes the ice between periods of each
game. The Taffy Abel Arena possesses two Zamboni machines.
www.lssu.edu/admissions/lakerterms.php

The vehicle used to prepare the rink's ice surface before the game,
and after each period. The Zamboni scrapes a thin layer off the ice,
heats the ice, and puts down a fresh layer of heated water that
freezes to form a new layer of ice.
www.seattlethunderbirds.com/fanzone/terms.html

A four-wheel drive vehicle that scrapes, clears and floods the surface
of a hockey rink, which was invented by a California rink operator
named Frank Zamboni. It grew out of an experiment he conducted with a
tractor pulling a sled across the ice at his rink.
www.cjonline.com/scarecrows/hockey/sca_glossary.shtml

the trademark for a machine that smooths the ice in an ice-skating rink 
wordnet.princeton.edu/perl/webwn

An ice resurfacer is a truck-like vehicle used to clean and smooth the
surface of an ice rink, originally developed by Frank J. Zamboni in
1949. The Zamboni Corporation and other companies manufacture ice
resurfacing machines, but because of the success of the Zamboni
Corporation, Zamboni has become a genericized trademark and most
people refer to any ice resurfacer as a Zamboni.
en.wikipedia.org/wiki/Zamboni