The basic issues is that the US Constitution requires one to be a "natural born citizen" in order to become President. It is now accepted that all people born on US soil are automatically considered "natural born citizens." Also, children of US citizens born overseas are likewise US citizens, except when the child's father was never resident in the US. The phrase "natural born" was removed from this particular law in 1795.
Obama's father was Kenyan, though he was a resident of the US for a while. Obama's mother was a US citizen. The basic birther argument is that Barack Obama was not actually born in the US. I'm not sure where all the claims come from. I think there is some obscure statement from his grandmother about him being born in Kenya. If he was not born on US soil, he is not a "natural born citizen" and would presumably be ineligible to be US President. (Incidentally, John McCain was born in the Panama canal zone under US control, so there was some noise during the election campaign as to whether he was a natural born citizen as well!)
The birther argument relies on two premises:
1) Barack Obama was not born on US soil.
2) US citizens born abroad do not qualify to be President under the "natural born citizen" clause.
The second premise is not particularly clear and would almost certainly go before the Supreme Court if it were to ever become an issue.
As far as the first premise, I've not been interested in digging through the dark underbelly of the internet conspiracy sites to look at all the claims. What I do know is that the birthers claim that Obama only produced a "Certification of Live Birth" and not a "Certificate of Live Birth." The later was an older form of the birth certificates that included the hospital where the child was born as well as the name of the physician. Hawaii no longer produces those "long form" birth certificates, so there is no alternate birth certificate that can be obtained. In addition, there was a notification of Mr. Obama's birth in the Honolulu Star-Bulletin on August 14, 1961:
- Click image to enlarge.

In order for the birther claims to be true, they have to believe that there has been a conspiracy in place since 1961 to fabricate evidence that a Kenyan was actually born in the US in order to get said Kenyan elected US President. Is it possible? Sure. Is it even remotely likely? Not a chance.