Under the law as in effect when Obama was born, a child with one non-citizen parent and one U.S. citizen parent would be a U.S. citizen at birth even if born abroad if the U.S. citizen parent had lived 10 years in the U.S., 5 of which were after the age of *14*; Obama’s mother was only 18 when Obama was born, so he would not have been a U.S. citizen at birth had he been born outside of the U.S.
The law was subsequently amended, in 1982 IIRC, and now a child born abroad with one alien parent and one citizen parent is a citizen at birth if the citizen parent lived in the U.S. for at least 5 years, of which at least 2 were after the age of 14. But Obama was born in 1961, not the 1980s, and thus it is the law in effect in 1961 that applied to his citizenship status at birth.
But if Obama was born in Hawaii as he claims (and as the photographed Certificate of Life Birth says), then he was a U.S. citizen at birth and thus a natural-born citizen eligible for the presidency.