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The Whomobile was a futuristic-looking vehicle designed by the Doctor, which was similar to a hovercraft and also capable of flight.
The Doctor first used the vehicle to search for the secret base of Operation Golden Age. (DW: Invasion of the Dinosaurs) It was later used to chase after Lupton when he stole the Doctor's Metebelis Crystal. (DW: Planet of the Spiders)
Behind the scenes
The Whomobile was never named in the show, and its name almost certainly would not be "the Whomobile" if it had been. The name clearly violates the fourth wall.
The Whomobile was personally commissioned by Jon Pertwee, who retained possession of it after his stint as the Doctor. It was capable of a speed of 105 mph.
When Pertwee demonstrated the vehicle on an episode of Blue Peter during his tenure, he revealed that the vehicle was road worthy and legal, though he mentioned that the BBC forbade him or any of the crew from removing it from the studios except for outside shoots.
The vehicle, which was also known as "Alien", was never given a name on screen; the name Whomobile was applied to it by Pertwee in interviews. It did not appear at all in the first story's original script or novelisation, both in which the Doctor used a motorbike. As the name is never uttered on screen, it doesn't violate the standard practice of never referring to the Doctor as "Doctor Who". However, the name was uttered in an audio drama featuring Pertwee and Elisabeth Sladen recorded for Glorious Goodwood, but which was never released until it appeared one of the Doctor Who at the BBC CD releases in the mid-2000s.
Pertwee later presented the Whomobile in the documentary Thirty Years in the TARDIS.