What is "bi-winning"?
I heard many people talking about this word, it seems to have something to do with an actor named Charlie Sheen. What exactly does this word mean in the context "I'm not bi-polar, I'm bi-winning!"?

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I'm bi-winning. I win here and I win there.

http://www.youtube.com/watch?v=J0NIMTPYYcU

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This explains the meaning of fail/win which is helpful in understanding why Charlie Sheen said he was bi-winning: http://www.youtube.com/watch?v=cilLKufYOhE

Charlie Sheen's erratic behavior has lead some to believe he may be bipolar. Bipolar disorder is a mental disorder in which the affected person has uncontrollable mood swings that cycle between an extreme high called mania and an extreme low called depression When he was accused of being bipolar in the interview, he said he was "bi-winning". By saying this he brushed off accusations of the disorder and continued to talk about how he's awesome or "full of win".

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LOL. That Charlie Sheen is just fantastic. To get the full joke, we need the context.

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It's a joke. He is making up his own word by copying the "bi" in bipolar.