When a child is born abroad to two U.S. citizen parents, the child is automatically a U.S. citizen at birth.
When a child (a legitimate or "legitimated" one) is born abroad to one U.S. citizen parent, that U.S. citizen parent must have spent sufficient time (as set forth by Congress) in the U.S.A. in order to transmit U.S. citizenship to his/her child at birth. For instance, if a U.S. citizen and his Greek wife have a child outside of the U.S., the U.S. citizen must have spent five years in the U.S. (two years after age 14) in order to be able to transmit U.S. citizenship to his child upon its birth. If the U.S. citizen does not meet the transmission requirements, his child is not a U.S. citizen at all; he can remedy that situation by filing an immigrant petition for the child, and the child will become a U.S. citizen upon being admitted to the U.S.A. with an immigrant visa. The child will still never be a "natural born" U.S. citizen.
Barack Obama was born to just one U.S. citizen, his mother, Ann Dunham; his father was a citizen of Kenya. In 1961, U.S. citizenship law said that a U.S. citizen must have spent ten years in the U.S.A., five years after age 14, in order to transmit U.S. citizenship to a child born abroad. See 7 FAM 1133.2-2 (in the State Department's Foreign Affairs Manual volume on citizenship). Ann Dunham gave birth to Obama when she was only 18 years old, so she did not meet the requirement that she live in the U.S.A. for five years after age 14. Under the law, she could not transmit U.S. citizenship to Barack at birth; the only way for him to be a U.S. citizen at birth was for him to be born in Hawaii.
Was President Obama born in Hawaii? I think so. There's a Hawaii birth certificate that says so. There are also two contemporaneous newspaper announcements of his birth in Hawaii. Those are sufficient evidence to me, and I accept that Obama was born in Hawaii.
However, I don't think that the evidence of his birth in Hawaii is so overwhelming as to make those who doubt his birth in Hawaii "crazy". Based on my experiences as a consular officer, I know that birth certificates are sometimes based on fraudulent information provided to a government office. Ann Dunham was in a very odd position when Obama was born; she was, unknowingly by her own report, in a bigamist marriage with a man whom she resided apart from within 2 months of her son's birth. If she, for some weird reason, had gone abroad while heavily pregnant and ended up giving birth outside the U.S.A., she would have had several reasons to pretend that her son was actually born in the U.S.A. (e.g., avoiding immigration paperwork, automatic U.S. citizenship for her son, and needing to get back to the U.S. soon in order to go to college in Washington state). Ann Dunham and her second husband don't appear to have been strictly honest in all of her son's paperwork; Obama's Indonesia school records claim that he had his stepfather's surname and was an Indonesian citizen. Since Ann Dunham appears to be a less than completely reliable source of information and the issue of Obama's birthplace is so central to his eligibility to be the U.S. president, it would really help if there were some baby pictures of Obama somewhere to show that he was a newborn in Hawaii. Did people not take baby pictures in 1961? Can't Obama show one of himself as a cute little wrinkly baby with his grandparents (who lived in Hawaii)? I would love this controversy to be put completely to rest, and such a photo would go a long way towards doing that.
UPDATE: Soon after I posted this entry, the "long form" birth certificate was released. It showed that Obama's birth was registered promptly and that he was born in a hospital. In light of this, I would revise the paragraph above in that now I would wonder whether a person might indeed have a brain dysfunction if still questioning whether Obama was born in Hawaii. But I'm not a psychiatrist, so I won't play one...even on the internet.