A gravity assist maneuver is when a spacecraft uses a planet to boost (or lower, but this is less common) its speed. To get a gravity assist, the orbit of the spacecraft is timed such that it passes near the planet with just the right velocity. The planet then pulls the spacecraft along in the direction of its orbit for as long as the spacecraft is near (if the planet weren't moving, it would only change the direction of the craft, not the magnitude of its velocity). After the momentum of the spacecraft carries it past the planet, it gets to keep the "free" velocity. The timing and velocity must be very precise, of course, if you want to be headed in just the right direction afterwards.
The Cassini-Huygens spacecraft is the record holder in gravity assists. It took two assists from Venus, one from Earth and another from Jupiter in order to get enough velocity to reach Saturn. Other notable assists include Voyager 2's hat trick: it took assists from Jupiter, then Saturn and then Uranus in order to reach Neptune. Neither of the Voyager craft had the energy to leave the solar system on their own; it took a big kick from Jupiter to give them solar escape velocity.
Of course, there's no such thing as a free lunch; mother nature's books must balance. The "free" energy comes at the expense of the planet's orbital velocity. Any momentum transfered to the spacecraft is lost by the planet, slowing its orbital velocity infinitesimally.