The relevant Wikipedia articles are:
Kosovo
International recognition of Kosovo
Political status of Kosovo
There are three major points here:
Recognition by the other states
Although 97 states have recognized Kosovo independence, only 11 of them are G20 states. Some major states oppose recognition out of concern that Kosovo example may encourage separatist movements within their borders. E.g.,
Within the EU, key supporters of Kosovo's statehood include France[14] and Germany.[15] The strongest opponents to Kosovo's statehood within the EU include Spain[16][17] and Greece.[18] The Spanish non-recognition of Kosovo is linked to the Spanish government's opposition to the Basque and Catalan independence movements,[16] while the Greek non-recognition of Kosovo is linked to the Cyprus dispute and Greece's historic relationship to Serbia.[18]
Recognition by the UN
Kosovo is not recognized by Russia and China, which are among the 5 memebrs holding veto power in the UN Security Council:
A number of states expressed concern over the unilateral character of Kosovo's declaration, or explicitly announced that they would not recognise an independent Kosovo. The United Nations Security Council (UNSC) remains divided on this issue: of its five members with veto power, three (the United States, the United Kingdom, and France) have recognised the declaration of independence, while the People's Republic of China has expressed concern, urging the continuation of the previous negotiation framework. The Russian Federation has rejected the declaration and considers it illegal.[7] In May 2008, Russia, China, and India released a joint statement calling for new negotiations between Belgrade and Pristina.[8]
Recognition by Serbia
Finally, Serbia considers Kosovo as its own territory, and therefore have the ability to block its de facto independence, even if such independence was possible de jure. The current autonomous status of Kosovo is governed by Brussels Agreemen (2013) between Serbia and Kosovo.
Serbia main objection to Kosovo independence is that it would violate Serbia's territorial integrity, as guaranteed by the UN charter:
Serbia argues that Kosovo's independence would be a violation of Serbia's sovereignty and territorial integrity and therefore contrary to the UN Charter and principles of international law.
The UN resolution 1244 seems to confirm Serbian view, urging a political solution to the crisis:
United Nations Security Council Resolution 1244, passed in 1999, reaffirmed in its preamble the "commitment of all Member States to the sovereignty and territorial integrity of the Federal Republic of Yugoslavia" and authorised "an international civil presence in Kosovo in order to provide an interim administration for Kosovo under which the people of Kosovo can enjoy substantial autonomy within the Federal Republic of Yugoslavia", i.e. the United Nations Interim Administration Mission in Kosovo (UNMIK), with security provided by a NATO-led Kosovo Force (KFOR). The Resolution also authorised a process to determine Kosovo's final status. Concerning the latter, Annex 1 to the Resolution states that the "political solution to the Kosovo crisis" should take "full account of ... the principles of sovereignty and territorial integrity of the Federal Republic of Yugoslavia".
However NATO holds an opinion that Kosovo could be made independent without Serbian agreement:
A NATO report on the question of Kosovo's status states that a region may have the right of secession in the case of foreign occupation or if the region is a colony of another nation. The NATO report claims, while mentioning disagreement on the interpretation, that a third condition exists when "a people whose right to internal self-determination has been thoroughly violated by a Government that does not represent the people" and suggests Kosovo qualifies under this condition. It concludes, while there is a case in favor of a right to secession for Kosovo, it rejects the right to secession in cases where regions are open to democratic mechanisms.[70]
Remark
In this context it is worth looking into the Political status of Taiwan, which used to be representative of China in the UN and the UN security council, but was excluded from both, and replaced by the People's Republic of China. Since PRC follows one China policy, many countries (notably the US) do not officially consider it now as a country, although maintain de facto diplomatic, economic and military relationships with Taiwan.