A jellyfish's movement is down to circular muscles located on the inside of the bell -- the main part of the body shaped like the top of an umbrella. As the muscles contract, the bell closes in on itself and ejects water to propel the jellyfish forward. After contracting, the bell relaxes and regains its original shape.
This was replicated in the vehicle using commercially-available shape memory alloys (SMA) -- smart materials that "remember" their original shape -- wrapped in carbon nanotubes and coated with a platinum black powder.
The robot is powered by heat-producing chemical reactions between the oxygen and hydrogen in water and the platinum on its surface. The heat given off by these reactions is transferred to the artificial muscles of the robot, causing them to transform into different shapes.
Can you guess who sponsors this research? That's right, the U.S. Navy. Link -via Metafilter