One of the key plot devices in nearly every Star Trek episode and film is the ability of starships to travel at light speed and beyond. This ability comes thanks to a propulsion system known in the show as warp drive.
What is Warp Drive?
According to Einstein's theories on relativity, it takes an infinite amount of energy to accelerate an object with mass up to the speed of light. So, on the surface it would appear that spacecraft traveling at (or exceeding) the speed of light is strictly impossible.
However, our current understanding of physics does not preclude the possibility of space itself traveling at or beyond the speed of light. In fact, some theorists claim that in the early Universe space-time expanded faster than the speed of light for a time.
Taking advantage of this loophole, if you will, the concept of a warp drive uses massive amounts of energy (extracted from matter-antimatter annihilations in the "warp core" of the ship) to encapsulate the starship in a bubble that "warps" the area around it. Space-time behind the vessel is expanded, while the space-time continuum is compressed in front. The net result is that the ship is pushed along as space-time expands and contracts around it.
In this way, the starship is effectively stationary relative to a local area of space-time and it is rather the fabric of the Universe that moves, carrying the starship along with it. (A happy byproduct of this is that the starship can get around such undesirable effects as time dilation and massive acceleration effects on the human body, which would reek havoc on science fiction story lines.)
This is different than traveling across the Universe using wormholes, theoretical structures that allow spaceships to travel from one point to another by tunneling through hyperspace -- effective taking a shortcut, since the ship remains bound to normal space-time.
Could We Someday Have Warp Drive?
There is nothing in our current understanding of theoretical physics that prohibits a warp type drive from being developed. However, it is still highly speculative.
In order to create and sustain a warp bubble (which in of itself is a challenge without also destroying the ship) a theoretical type of matter would have to exist with negative mass. Essentially the negative root of a quadratic equation that arises in relativity, theorists have argued whether this solution has physical meaning. In short, we don't even know if negative mass (and henceforth negative energy) exists anywhere in the Universe.
Having never detected negative mass particles, their existence remains an unanswered question. Supposing, however, that such matter did exist one could devise such a drive system. In fact, at least one such design has garnered attention in the Alcubierre drive.
In this iteration of the warp drive the starship would ride a "wave" of space time, much like a surfer rides a wave on the ocean. But just because a drive system could be theoretically possible, does not mean that it is pragmatic. The sheer amount of energy required to create the necessary expansion and contraction of space-time would exceed the output of the Sun.
Even with a power source as potent as the one described in the Star Trek series, it seems highly implausible that such travel will ever be possible. At the very least, we don't have an evolved enough understanding about the physical nature and composition of the Universe to really evaluate what is possible in the realm of faster than light travel.