Magenetic Resonance Imaging utilizes nuclear magnetic resonance to
produce detailed images of the interior of the human body.  A fairly
detailed discussion of the physics of how this works can be found at
http://www.cis.rit.edu/htbooks/mri/bmri.htm

The relevant issue for tattoo enthusiasts is that MRI utilizes a
strong magnetic field and radio-frequency radiation which can interact 
with some tattoo inks containing metal salts.  Several people have
reported some mild discomfort during MRI.  This took the form of heat
in the tattooed areas.  The treatment for this was to apply cold
compresses to the areas to absorb the heat.  Apparently, this does not 
affect the quality of the images recorded by the MRI.