A Dopp Kit is, for better or worse, better known as a toiletry kit in America.
From my limited research on the topic, it seems that the concept was initially invented by a Jerome Harris, who worked at his uncle's company. As this company was named the 'Charles Doppelt Company,' it is not too difficult to see how the trademarked name of this leather bag became 'Dopp kit,' as opposed to 'Harris kit.' These trademarked kits were issued to American GIs during World War II and thus became relatively well known; however, even though I was born and raised in America, I can't really say that I had ever heard of these before. Dopp kits are still available through various retailers.
The Dopp company and trademark is now owned by Samsonite.
Sources:
http://www.quinion.com/words/qa/qa-dop1.htm
http://www.word-detective.com/103001.html