"Dupe" is an abbreviation for "duplicate." In gaming terms, this means to create a usable copy of an original item, either for personal use or trade. It's essentially counterfeiting, and while this is largely harmless (like cheating at solitaire) in single player games, it's a major issue in multiplayer online games. MMORPGs have carefully balanced economies, where the scarcity of items is used to balance their utility, value, and demand. when players find ways to cheat by creating dupes, the economy and game difficulty is usually thrown totally out of whack.
There was constant duping of every item in Diablo I, since the game code had a bug that allowed players to easily duplicate any item.
Duping was more difficult in Diablo II since Blizzard North programmed the game to prevent it, and used the client/server technology on Battle.net to keep players from housing the game data on their machines. Techniques to dupe were discovered anyway, with early outbreaks of duped SOJs. Duping became a bigger factor in Diablo II: Lord of Destruction, when ultra rare runes were introduced, and high level runewords were added that made duping those runes enormously valuable.
The D3 Team has said they will be able to control and eliminate duping in Diablo III, thanks to the improved client/server technology they've perfected during their years running World of Warcraft. Most players certainly hope they're right, but no game ever has perfect security.
Duping Drawbacks
Individual players often enjoy duping or duped items, since it gives them great items they wouldn't otherwise have had. Aside from the moral issues of cheating, the problem when duping becomes widespread is that causes mudflation and breaks the game economy and difficulty. If super quality items are duped and become very common, that lowers the value of all other items, and makes regular gameplay worthless, in terms of item-hunting. This breaks the game economy since players can no longer expect to find items worth trading. Also, the fact that players are using very high quality duped items breaks the game difficulty, since the designers balance the game based on reasonable expectations of the quality of equipment players will have. If players all suddenly have nothing but uber gear on, the monsters will be too easy for them, and if the designers turn up the monster difficult, the game becomes too hard for new players or players who aren't using the duped items.