Memory Alpha, the wiki for all things Star Trek, describes the Enterprise's warp drive thusly:
Warp drive is a technology that allows space travel at faster-than-light speeds. It does this by generating warp fields to form a subspace bubble that envelops the starship, distorting the local spacetime continuum and moving the starship at velocities that exceed the speed of light. These velocities are referred to as warp factors.
Two physicists at Baylor University are proposing that string theory suggests that faster-than-light travel using something like the Enterprise's warp drive may be possible.
[They] theorize that by manipulating the extra spatial dimensions of string theory around a spaceship with an extremely large amount of energy, it would create a "bubble" that could cause the ship to travel faster than the speed of light….
The method is based on the Alcubierre drive, which proposes expanding the fabric of space behind a ship and shrinking space-time in front of the ship. The ship would not actually move, rather the ship would sit in a bubble between the expanding and shrinking space-time dimensions. Since space would move around the ship, the theory does not violate Einstein's Theory of Relativity, which states that it would take an infinite amount of energy to accelerate a massive object to the speed of light.
The Baylor physicists estimate that the amount of energy needed to influence the extra dimension is equivalent to the entire mass of Jupiter being converted into pure energy for a ship measuring roughly 10 meters by 10 meters by 10 meters.
The Baylor press release on faster-than-light space travel here.