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"Science fiction writers, I am sorry to say, really do not know anything. We can't talk about science, because our knowledge of it is limited and unofficial, and usually our fiction is dreadful."
- Philip K. Dick

Diamagnetic Levitation  
  Levitation accomplished using magnetic field's interference with the motion of electrons orbiting the atoms or molecules of a material.  

Diamagnetism has been known since the early 19th century; the first experimental confirmation of diamagnetic levitation was done with bismuth in 1939. The technique also works with living creatures; a living frog was successfully levitated when placed in the bore of a Bitter solenoid with a field strength of about 16 Tesla (the magnetic field strength of the Earth varies from about .3 to .6 gauss at sea level; one Tesla equals 10,000 gauss).

The ship rolled out of the hangar like an air-borne whale, moving slowly, its diamagnetized hull clearing the smooth-packed clay of the field by three inches. The ship came to a halt, pinpointed at the lip of a take-off pit. The diamagnetic field strengthened progressively towards the ship's prow and it began tipping upward. Terens was mercifully unaware of this as the pilot room turned on its universal gimbals to meet the shifting gravity. Majestically, the ship's rear flanges fitted into the appropriate grooves of the pit. It stood upright, pointing to the sky.
From The Currents of Space, by Isaac Asimov.
Published by Unknown in 1962
Additional resources -

Diamagnetism occurs as a result of a magnetic field's interference with the motion of electrons orbiting the atoms or molecules of a material. When matter is placed in a magnetic field, the magnetic force acts upon the moving electrons in the matter, causing the electrons to be deflected. This movement of the electrons interferes with the motion of the magnetic field, so the atoms internally oppose the field. This causes the material to be slightly repelled by the magnetic field.

All materials demonstrate some degree of diamagnetism. Magnetic levitation occurs when the force on such an object is strong enough to balance the weight of the object itself. The technique of immersing an object in a liquid to enhance the diamagnetic effect is called "magneto-Archimedes" levitation, due to the buoyancy provided by the liquid.

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Additional resources:
  More Ideas and Technology from The Currents of Space
  More Ideas and Technology by Isaac Asimov
  Tech news articles related to The Currents of Space
  Tech news articles related to works by Isaac Asimov

Diamagnetic Levitation-related news articles:
  - Scientists Succeed At (Cryogenically Enhanced Magneto-Archimedes) Levitation

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