Is the actual quote, not “Houston, we have a problem”, as it is more commonly known as.  The famous Apollo 13 ‘successful failure’ happened in 1970, the mission that was intended to be the third try to land on the moon.  The Odyssey (the command module) and its Aquarius (the lunar module) launched as the spacecraft and headed for the moon, specifically the Fra Mauro highlands.  The mission was aborted when the oxygen tank that was located in the service module exploded and left the crew, consisting of Jim Lovell, Jack Swigert, and Fred Haise, under significant amount of pressure. 

            “We’re venting something out into the- into space.”-James Lovell Jr.

The ground controllers did not have a fun time, either, and neither did the public as they breathlessly watched Apollo 13 try to make its way back home.
It was critical for the survival of the crew to use the Aquarius (which was still attached to the Odyssey) as a ‘lifeboat’ (the explosion had damaged the spacecraft’s electrical systems, which made it impossible to keep up the generation of power to maintain the Odyssey operational).  The ground controllers and the crew made a frantic calculation to correct their course from the original one to the moon in order to have the spacecraft go on a free return trajectory.  This feat was accomplished by firing the Aquarius’s descent engine twice, leading the spacecraft to accelerate (after looping around the moon) its return to Earth.
The crew did not sleep as they undocked and jettisoned the Aquarius, retaining it for their escape back home to Earth, and releasing the damaged service module.  The mission is named the ‘successful failure’ because the crew did make it safely back to Earth despite all their troubles.

*Pictures from http://en.wikipedia.org/wiki/Apollo_13

Sources:
Haukap, Anthony W. Project: Apollo 13 Ex Luna, Scientia (From the Moon, Knowledge). 12 Oct 2003. <http://209.145.176.7/~090/awh/as13.html#tv2>
(primary)

Moritz, Kylie. Apollo 13 History. 13 Oct 2003. <http://spaceflight.nasa.gov/history/apollo/apollo13/>

Prasad, Sivram. Apollo 13 Review Board (Cortright Commission). 13 Oct 2003. <http://history.nasa.gov/ap13rb/ch3.pdf > (primary)

Wikipedia. Apollo 13. 13 Oct 2003. <http://en.wikipedia.org/wiki/Apollo_13>

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