Warning: There is no way to decouple politics from this; I have tried to be objective.
What ended as the ARM (Asteroid Redirect Mission) officially began in a speech by then-President Obama in April 2010 at Kennedy Space Center. This was the speech in which President Obama discussed the cancellation of the Constellation lunar landing program:
"But I just have to say pretty bluntly here. We’ve been there
The Obama administration's February 2010 budget request had already proposed the project cancellation, so this was not an announcement of that. In fact, that budget had proposed cancelling the Orion spacecraft as well. However, in this speech President Obama made an announcement about a crewed mission to an asteroid by 2025 which would use Orion. Please note the exact wording - that is important.
“And by 2025 we expect new spacecraft designed for long journeys to
allow us to begin the first ever crewed missions beyond the Moon into
deep space. So we’ll start by sending astronauts to an asteroid for
the first time in history.”
Note: this type of mission had been studied before as part of Constellation planning as this November 2006 article and this November 2009 article describe, so it was not completely novel.
In June 2011 NASA was still trying to plan the mission. The thought was to send a crew to a near-Earth asteroid but no destination had been chosen.
In 2013 and March 2014 President Obama's NASA Administrator Bolden was talking in terms of capturing a small asteroid whole and returning it to an orbit near Earth. (To meet the "sending astronauts to an asteroid" goal literally)
By May 2015 both ideas of sending a crew to a deep-space destination or moving an asteroid to a Earth orbit had been abandoned in favor of what was really a sample return mission (although that term was never officially used). This would involve a robotic spacecraft which obtained a sample from an asteroid and returned it to an orbit near Earth, from which a crew would retrieve it.
By June 2017 the project was cancelled by the Trump administration.
To sum up, a sad history of descoping followed by cancellation.
- Send crew to an asteroid (April 2010)
- Bring an asteroid back (April 2013)
- Bring a rock back (May 2015)
- Cancellation (June 2017)