That's a very interesting question. Part of the answer is already included in the question itself - for it is not evident at all that the question should not start with the word "what".
As was pointed out -- correctly I believe -- a determination of the validity of a mathematical proof is a social process. This may come somewhat as a surprise -- especially for high priests of mathematics who believe with all their heart that mathematics equals truth.
The mathematical community is first and foremost a community. It has its institutions, governments, ambassadors, pundits, enthusiasts, cults and rebellious underground movements. The current state of affairs was concisely summarized by Clement Guarin in the comments above. Observe how structured is the social process -- first the proof inventors have to believe in their proof. Then they must present it in front of other people. These other people can be members of their faculty, or readers on a site like this one. Then they ought to present it yet in front of still other people -- presumably holding more distinguished posts than the first ones -- these could be journal editors or otherwise distinguished persona in widely accepted forums, such as some mathematicians who post on the real-mathematics sites. The purpose of this all is to expose the proof to as many eyes as possible, in the hope that if there were some mistake somewhere, it surely would have been found. But this has not always been the case. See the story of the Busemann-Petty problem.