You want the truth about the whole "birther" thing? Listen closely because there's a test on this later. (Lol, just kidding.)
In a nutshell, "birthers" as they are called, claim President Barack Obama is not a natural-born American citizen, which would make him ineligible for the position he currently occupies. The name "birther" comes from the fact that they believe his birth certificate to be a forgery, and that his real birth certificate (which he is supposedly hiding) proves that he was born in Kenya/Indonesia/wherever. This rumor was started during the 2008 campaign by, ironically, some of the angrier supporters of Hillary Clinton (not Clinton herself, as some claim), who were upset that she was losing the Democratic primary to Obama, and was picked up by many on the far-right fringe after Obama was elected.
The thing is, this theory, like most conspiracy theories, is absurd when actual logic is applied. In order for such a claim to work, it would require either a) planning by "Them" going back half a century (Honolulu newspapers reported his birth), in a time when the idea of a non-white President was laughable, as well as a lot of natural charm, brilliance and political success on the part of their supposed puppet to actually get elected 47 years later, or b) Obama not only successfully forging his long-form birth certificate, but also fabricating those newspaper announcements, fabricating his college records, bribing immigration officials, and bribing or fooling election officials. (Wouldn't someone capable of that simply bribe a few Republican Representatives to make sure such nonsense stopped?)
For the record, under current laws, anyone born to an American citizen is also a natural-born American citizen, no matter where this occurred. The President was born before this law was implemented, but even under the statutes he was born under, he only needs to spend a few years in the United States to claim natural-born citizenship; his academic record alone validates that claim. Some claim that being a natural-born citizen isn't enough — you must actually be born in the US to run for President. This is in spite of the fact that the Constitution does not make such a requirement, only requiring a) either natural-born citizen status or being a citizen of the US at the time the constitution was ratified (the latter doesn't apply anymore, but it's why George Washington was allowed to be president despite not being a natural born citizen), b) having lived in the US for fourteen years, and c) being at least 35 years old. Even if there were a "born on US soil" requirement, there is ample evidence confirming that Obama was born in Hawaii. (This, by the way, is why Ted Cruz is, indeed eligible to be President.)
One variation claims that, since Obama's father (a native of British Kenya) held British citizenship at the time of his son's birth, then Obama holds dual US and British citizenship, which they feel would make him ineligible even if he was a natural-born citizen. However, there is nothing in the Constitution saying that dual citizenship makes a candidate ineligible, and even if there was, the 14th Amendment to the Constitution would override it. Other claims allege that Obama's mother had renounced her US citizenship and, by extension, her son's (she had not, and even if she did it wouldn't have affected Obama's status), that a trip by Obama to Pakistan in 1981 could only have been accomplished with a foreign passport due to an alleged ban on travel between the US and Pakistan (no such ban existed), and that Obama's enrollment in a school in Indonesia in his youth could only have been done if he wasn't a US citizen at the time (living outside the US as a minor, as ruled in Perkins v. Elg, does not cause you to lose your citizenship).
Another variation claims that Obama being born in Hawaii means that he's not a natural-born citizen, the argument being that Hawaii is not legally part of the US, but rather, is an independent kingdom under US occupation — and therefore, nobody who was born there can claim natural-born US citizenship unless they meet one of the other requirements for it. This is only believed by people who insist that Virginia is not truly a state because it's a "commonwealth". (In other words, it's absurd.)
Ironically, Obama's 2008 opponent, John McCain, was born in the Panama Canal Zone (or Panama itself, some believe), meaning that, by some variants on this theory, he too is ineligible to be President. Of course, this ignores the fact that his father was a serving US naval officer, stationed in Panama, which means that McCain automatically is a natural-born citizen (though retroactively; the laws allowing for this were only passed after he was born). Oddly, there was no question of his eligibility during the 2008 campaign.
Now, when the Birthers produced a birth certificate of their own claiming that Obama had been born in Kenya, it was found to not only be a forgery, but a very unconvincing forgery, due to several instances of shoddy research. For one, it listed his birthplace as "Mombasa, Republic of Kenya", even though a) in 1961, the year of Obama's birth, Mombasa was a part of Zanzibar, not Kenya, and b) Kenya was still a British colony at the time, not a republic. In addition, the format of the certificate bore no resemblance to that used in Kenya at the time. It was shown to be an altered version of a birth certificate issued in South Australia in 1959.
In short, I suggest not listening to the President's detractors. (Demands for his impeachment started even before his inauguration, before he could have even done any Presidential duties that would have warranted it.) Most have no idea what they're talking about.
Hell, most of Bush's detractors have no idea what they're talking about! Amnesty International's demands that he and Dick Cheney be arrested and tried by the ICJ at the Hague are all utterly pointless, given that the ICJ can only hear a case if both parties to the dispute agree to their mediation. No one ever does the research before making such foolish demands and claims.