Diplomacy is the art of making relations with other game entities in Civilization V. The world is huge and filled with other civilizations whose leaders are at least as cunning and determined as you are. Some are honest and others are liars; some are warlike and others prefer peace. But all want to win.
You can accomplish a lot through diplomacy: you can trade to make profit off your nation's excess production; you can gain allies and isolate your enemies; you can create defensive and offensive pacts; you can advance your technology through cooperative research ventures; you can end wars that are going badly for you; you can bluff the credulous and bully the timid.
And finally, if you make enough allies, you can achieve a Diplomatic Victory!