A euphemism for defecating, dropping a load, etc. You get the idea. I hadn't really thought about where this came from until it was requested that I node it. A somewhat older euphemism for this particular activity is "going number 2". "Deuce" is a slangy term for "two", and hence dropping the deuce is dropping a number 2, or for those who are a bit slow...

Taking a shit.

While, there you have it. My apologies.