The diarrhea (also spelled diarrhoea) that is suffered by tourists when travelling to foreign parts, in this case Mexico.

Montezuma was Emperor of Mexico, 1502-20. The sickness, more formally called "traveller's diarrhea", is usually caused by drinking unsterilised water or eating spicy food that visitors aren't accustomed to. It is a bacteriological illness, always uncomfortable, and occasionally serious. Most cases are caused by the enterotoxigenic E. coli bacterium.

The revenge element of the phrase alludes to countries that were previously colonized by stronger countries and are now, in this small way, getting their own back.

Names for it vary depending on the part of the world concerned. These usually have comic names, which reflect the embarrassment felt by the sufferer and the amusement in the lucky non-sufferers. Montezuma's Revenge is also known as the Gringo Gallop and the Aztec Two-step.

Those unlucky enough to suffer from it in Asia will hear it called Ghandi's Revenge, Gyppy Tummy, Delhi Belly, the Rangoon Runs, Tokyo Trots etc.

Gyppy Tummy goes back to at least the Second World War, but Montezuma's Revenge isn't particularly old. The earliest citation of it in print is from The Washington Post, January 1960:

"‘Wasn't there anything you didn't like?’ and they admit that, like most Americans, they suffered a three-day gastric upset described by a variety of names, names like the Gringo Gallop and Montezuma's Revenge."