A warp drive is a theoretical type of space propulsion drive which would allow spacecraft to travel faster than light by somehow warping or distorting spacetime around them. Science fiction writers, and certain futurism-oriented physicists, envision the warp drive as an important building block for interstellar travel, since in ordinary space an object cannot travel faster than the speed of light, and therefore cannot traverse the distances between stars except in trips lasting from, in the very best-case scenario, several years (to the nearest star, Alpha Centauri), tens of thousands of years (to cross the Milky Way Galaxy), or millions of years (to reach the nearest galaxy to our own, Andromeda).
In ordinary spacetime, an object cannot exceed the speed of light, and as it approaches the speed of light, it approaches infinite mass. This means, in practice, that a fast spacecraft can only travel a fraction of the speed of light - but even if we were to build an extraordinarily powerful craft (far beyond our current means), it would still only travel bleow the speed of light. This does not make travel between the stars impossible, but it does mean that a single trip will take an unrealistically long time. For this reason, a drive capable of propelling a ship to more than the speed of light is the holy grail of space travel: it would make interstellar travel not just theoretically possible, but practically doable for individual human beings.
A warp drive is a type of theoretical faster-than-light drive which achieves this trick not simply by accelerating past the speed of light with raw power (which is not possible), but by somehow altering the basic rules of spacetime in its immediate vicinity so that it can travel at a speed ordinarily impossible, relative to the origin and destination points of the trip. It is important to emphasize that a faster-than-light drive, like a warp drive, would only create a speed apparently faster than light to an obsever. The spacecraft could not travel faster than light at any given time, but it could travel through space in such a way that it reached its destination faster than light would.
One popular way to do this is via a controlled wormhole, although this is generally considered to be impossible. Another is the warp drive, made famous by Star Trek, but which also exists in theoretical form as what is known as the Alcubierre drive. According to Mexican physicist Miguel Alcubierre, such a drive could create apparently faster-than-light travel by distorting spacetime surrounding a spacecraft, creating a "bubble" or wave around the ship by somehow contracting space in front of it and thinning out or extending space behind it. The warp ship would then effectively "surf" this wave, riding along in a region of space which was itself moving seemingly faster than the speed of light through surrounding spacetime. At no time would the spacecraft itself travel at or beyond the speed of light within this bubble, but because of the effects of its warp bubble, it could travel at a speed faster than a beam of light outside the bubble. Alcubierre's theory bears mild resemblance to some later pseudo-astrophysical explanations of warp travel in the Star Trek fictional universe as well as in other, less-well known science fiction.
The Alcubierre drive does not violate basic principles of relativity, and does not do the impossible by actually suggesting that any spacecraft be propelled to faster-than-light speeds. Nevertheless, there are several vital problems which would seem to make this an impractical method of propulsion - not to mention entirely different than the sort of easy zipping about imagined in Star Trek.
First, the energy requirements for creating and sustaining the warp bubble are immense - so much so that, barring the discovery of some unimaginably powerful new energy source, the fuel for the spacecraft would have to be laid out on its path in advance. However, creating fuel stations along the route would require making the distance either with another warp vessel (which would in turn require its own fuel source), or by first sending a sub-light-speed spacecraft along to seed the route with fuel caches. The first of these is an impossibility (i.e. to make an Alcubierre drive work, you would first have to have another ship with an already working Alcubierre drive, which the first inventor would not).
The second is theoretically possible, but it means that even if the Alcubierre drive-equipped ship could make the trip faster than the speed of light, the total time necessary to make the voyage would actually be much less than the speed of light (since the seed ship would need spend years, decades, millennia, or millions of years to make the trip first, depending on the distance to be traversed). Still, this could theoretically be resolved, even if impractically: we could send out large numbers of unmanned fuel tankers, and then begin using Alcubierre warp ships as the routes became available (and also send out new tankers in the meantime to keep routes continually replenished).
The other problem is a more serious one. By definition, a ship within the warp bubble would appear to be travelling faster than an object in the surrounding space - in much the same way that a pilot in a supersonic aircraft is travelling faster than sound waves around him. How the ship would perceive and respond to conditions at or beyond the edge of its warp bubble is unclear - and in fact, it may be unable to do so. It is at least possible that the pilot of an Alcubierre warp ship would not be able to steer or stop his ship. Even if a ship could create a warp bubble, an entirely separate (and heretofore unknown) process would be needed to leave the bubble at the end of the trip.
It should be noted that Alcubierre originally proposed the warp drive as a helpful thought experiment for imagining how faster-than-light warp travel could work, not as a concrete proposal for the construction of a specific spacecraft. As such, it is helpful both in illuminating some of the possibilities, and also in illustrating some of the challenges, of building a faster-than-light spacecraft.