The dots sprinkled across the board are supply centers, and for every one of them that you own, every couple of turns you'll get to construct more armies so that your # of supply centers = your # of armies. Each nation starts with 3 supply centers and 3 armies, except for russia, who gets 4. Your goal by the end of the game is to own 18 supply centers, which means you automatically win.
The game plays somewhat like risk, but with each army being equal in strength. The only way to defeat someone else's army is to support yours with a second army next to it, whether it be your own or someone else's. Here is where the key to the game lies. Diplomacy is all about convincing your opponents to work with you against a common foe, then (probably) backstabbing them before they can do the same to you.
Here's a better breakdown of the rules in ~10 minutes.
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I'd like to get a game set up sometime this weekend, if any of you have played before or are willing to give it a go.