That's not the kind of show he means!! He's talking about a competition between fish.
Yes there are. All clubs have some kind of shows, and bigger clubs have regional and national shows, just like for cats and dogs.
Some websites I could find quickly...
http://www.cichlid.org/ShowSanctioning.html
http://www.rmkc.org/KoiShow.htm
Most local clubs don't do a lot of publicity outside their area, and since it's just changed to 2009, a lot of the websites don't have current information.
Look to see what aquarium clubs are in your area, and contact them for show dates. Or find a national or international club based on the kinds of fish you like. Their shows are more spectacular because they have more entries!
http://www.aquariumhobbyist.com/society.html - some clubs
Also you can find a club listing in the back of most fish magazines at the pet stores.