A birther is someone who questions whether Barack Obama was a natural-born citizen of the U.S., a constitutional requirement to be President.
On June 12, 2008, Obama posted an image of the "short form" of his Hawaiian birth certificate on the internet.(1)
Birthers argued that the "short form" wasn't equivalent to a "birth certificate" even though the State of Hawaii stated that "the short form" is "prima facie evidence of the fact of birth in any court proceeding".(1)
In 2011, the State of Hawaii said that the "long form" birth certificate was kept in archives and couldn't be released to anybody, including President Obama.(1)
On March 2011, Trump became one of the more prominent "birthers".(1)
On March 30, 2011, Trump told Bill O’Reilly, "Obama doesn't have a birth certificate. Now, he may have one but there is something on that birth certificate — maybe religion, maybe it says he's a Muslim, I don't know." Even though O'Reilly had told Trump earlier that his team had turned up two Obama birth announcements in local Hawaiian newspapers.
On April 7, 2011, Trump told NBC, "You are not allowed to be a president if you're not born in this country. Right now, I have real doubts." "I have people [in Hawaii] that have been studying it and they cannot believe what they're finding."(2)
8/25/12: Even if Obama were born in Kenya , the president still would have been a U.S. citizen at birth, because his mother was an American, a fact which nobody to this date denies. http://www.huffingtonpost.com/michael-russnow/obama-birthers
On April 21, 2011, Trump gave CNN an update on his investigation, "Americans will be "very surprised" by what I've found. "We're looking into it very, very strongly. At a certain point in time I'll be revealing some interesting things."(2)
On April 22, 2011, Obama requested Hawaii waive normal department policy and provide him with certified copies of the "long form" birth certificate, which Hawaii did on April 25, 2011.(1)
On April 27, 2011, an image of the "long form" was posted on the White House website.(1)
On May 1, 2011, Obama released his birth video at the annual White House Correspondents dinner and said, "Now, Trump can focus on more important matters like, did we fake the moon landing?"
Birthers claimed that the "long form" was a forgery due to anachronistic elements and to existence of layers in the PDP file. All these claims are debunked by the article linked below:
On July 9, 2015, CNN's Anderson Cooper asked Trump whether Trump thought Obama was born in the U.S. and Trump responded, “I don’t know. I really don’t know. I don’t know why he wouldn’t release his records.” “I was able to get something. I don’t know what the hell it was, but it doesn’t matter. Because I’m off that subject." http://www.politico.com/2015/07/donald-trump-birther-obama