The UN's man in Kosovo, ex-Finnish President Martti Ahtisaari, has delivered a report to the Security Council saying the territory should become an independent country. Kosovo, a part of Serbia that's full of Albanians, has been run by the UN since the war in 1999. The plans would give it an internationally-supervised sovereignty. If all's well, Kosovo would get full independence after that. The Kosovo Albanians are happy enough with this. What actually happens next is up to Serbia and its old ally Russia.
Kosovo has a near-mythical place in Serb history as a cradle of Serb nationalism, and they desperately don't want to let it go. In 1989, Slobodan Milosevic's blood-and-thunder rhetoric on the anniversary of the 1389 Battle of Kosovo helped stoke anti-Muslim feeling among Serbs, which eventually led to the collapse of Yugoslavia and the wars of the mid-nineties. Later, he tried to kick all the Albanians out of Kosovo, which led to the war with NATO, which ironically looks like guaranteeing that Kosovo won't be part of Serbia anymore.
Losing Kosovo would make any Serb politician very unpopular indeed, especially as most of them were recently re-elected because of their firm opposition to exactly that. But the Serb politicians also know there's little chance of doing all the things needed to make their country better again (such as starting talks on EU entry) unless the Kosovo problem is sorted out. They're trying to stall things as long as they can, in the hope the UN agrees to an autonomous Kosovo that stops short of full independence.
This is where the Russians come in. They're already calling for more talks, and by threatening to wield their veto in the Security Council, they might make sure that happens. The Russians are doing this partly because they feel a sense of allegiance to their old friends in Belgrade, but mainly because President Putin is set on trying to keep Russia's sphere of influence in place. It's another example of the Kremlin setting itself up as a challenger to the west, rather than a friend.