One of the unique things about Diplomacy is the ease with which it can be played by mail (or email, nowadays). That's not to say it's not a terrific (if somewhat long) game to play across a tabletop, but PBM Diplomacy was a staple for many gamers back in the hoary mists of antiquity. In fact, Gary Gygax was a contributor to several such diplomacy 'zines, and some of the earliest non-TSR D&D articles (or mentions) are to be found in such places, such as his Expedition to the Black Reservoir (published in El Conquistator in 1975) and Len Lakofka's own Diplomacy 'zine Liaisons Dangereuses, which lasted from 1969 to 1977.
The game portrays the power politics in Europe at the start of the 20th century. Seven powers-- England, France, Italy, Germany, Austria-Hungary, Russia, and Turkey-- are vying for dominance. Players have armies and fleets, and write down movement orders, which are then revealed simultaneously. Combat occurs if units from different nations attempt to occupy the same territory, in a very simple system. Basically, if you have more units supporting your move, you get to occupy the territory. Units are only destroyed if they can't retreat or if a country doesn't have enough supply territories to support them all.
Where Diplomacy really shines, though, is in the behind-the-scenes negotiation between the players. If you promise to move your fleet out of Marsailles, I'll give you the supply center in Portugal, so we can stop Turkey from moving his fleet into the central Mediterranean... etc. etc. etc. The joy comes from making complicated deals like that and then back-stabbing your would-be ally in order to grab all the supply centers for yourself.
It's a remarkably well-balanced game (although some claim that the two "Wicked Witches", England and Turkey, do better than others such as Italy or Austria-Hungary) and the mechanics are so easy that they can be taught in 5 minutes. I've heard complaints that it's too long a game (4+ hours), but that can be alleviated by simply clamping down on the amount of time players can spend on diplomacy between turns. And besides, when you have a game like this that not only encourages, but requires, players to interact with one another, the day becomes less about the game itself and more about the time you're spending with friends.
And that's one of the reasons it's one of the games I love.