According to the citizenship rules in place at the time of Obama's election (the rules are applied retroactively provided they are less restrictive, but not if they are more restrictive - the new rules are less restrictive and actually remove the illegal sexism in the system) Obama would be a natural born citizen due to his mother being one.
At the time of the election, to be a natural born citizen, there were many ways - be born here; be born on an official US installation (such as an embassy, a military base, etc) or be born while your citizen parent was on official US government duty abroad; and then there is the parentage path which I will expound on.
Assume Obama was born in the pleasure palaces of the Ayatollah of Iran just for the sake of argument...can he be a natural born citizen still? Yes, provided his mother is a natural born citizen and meets the other requirements. At the time of Obama's election, the requirement was that she be a natural born citizen and that she lived at least 5 of the last 10 years after he 14th birthday in the US (or its official areas as stated above).
Was she a US citizen? Yes, she was born an a military base in the US. Cannot get any more American than that. Did she live 5 of the last 10 years after her 14th birthday in the US? That is where we have a potential legal fight.
Barack was born when his mother was only 18 - and that means she only lived in the US for 4 years of the last 10 since her 14th birthday and not 5. She lived ALL of her years in the US, but not 5. Why not 5? For the math impaired, it is impossible to live 5 years between the ages of 14 and 18 due to there only being 4 years between those ages.
They could legally petition the SCOTUS to rule Barack a non-US citizen at all (due to never going through the citizenship path) at the time of his first election due to that technicality. The letter of the law is on their side, but the spirit of the law says that since she lived 100% of the time after she turned 14 inside the US that she passed on citizenship to her child.
The purpose of the amount of time living in the US is two fold. The founders were afraid citizens would move to France (for example) and live there for 5 generations, giving birth to US citizens who have never set foot in the US and whose great grandparents did not either. We would have foreigners as US citizens. The other reason is what would happen if a pregnant women went to the Canadian side of Niagra Falls, slipped, and went into early labor. Her child would be born in Canada...does that mean the child is not a US citizen? No, the rules say the child is still a US citizen, which is a good thing.
Thus, it does not matter if Barack was born in the US or not, his mother passed on citizenship to him at birth, which makes him a natural born citizen. There is a very small possibility the SCOTUS will rule on the technicality and claim he was not a natural born citizen at the time of his first election, but I sincerely doubt it.