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Blon Slitheen intended to use this to escape early 21st century Cardiff, blowing up Earth and riding away on the subsequent shockwave of energy. After they realised her plan and captured her, the Ninth Doctor and Jack Harkness confiscated it and connected it to augment the recharging of the Doctor's TARDIS. Jack was curious to know where Blon had gotten her hands on it, knowing that the technology was far beyond what her people were capable of. (TV: Boom Town) Finding Jack in the TARDIS struggling with the device, River Song called it a rift extrapolator, and helped him reconfigure it. River fully charged the TARDIS, and the two shared a kiss before an earthquake interrupted. (AUDIO: R&J)
This was the Cardiff Space-Time Rift opening up. Knowing that anyone who would arrest her would have confiscated the extrapolator, Blon had programmed the board to connect the nearest alien power source — in this instance, the TARDIS — to open the Rift. The strain of pulling the ship apart opened the Heart of the TARDIS and regressed Blon to an egg. The extrapolator remained connected to TARDIS systems. (TV: Boom Town)
The extrapolator was later used by the Ninth Doctor and Jack Harkness to create a force field around the TARDIS to protect it from the Daleks' missiles. It was again used to shield the upper floors of the Game Station from the Daleks' transmats. (TV: The Parting of the Ways)
By the time the Tenth Doctor used it to move the TARDIS two hundred feet away from the Empress of the Racnoss, it appeared to have been integrated into the TARDIS systems. It was connected by a line and wrapped in TARDIS coral. The Doctor encouraged operation with percussive maintenance. (TV: The Runaway Bride) Later, the extrapolator's shielding was still in use by the Doctor, but it was powerless against a fully-formed Dalek empire capable of taking planets out of orbit. (TV: Journey's End)
Its ultimate fate is unknown, but was likely destroyed by the Tenth Doctor's violent regeneration, and/or the TARDIS' subsequent crash landing in Leadworth.