Space Elevator Explained (Video)
Here is an interesting clip explaining how one of the many possible space elevator technologies, the space tether, would work. Please note that this space elevator design is currently the only one that is the subject of active research and commercial interest in space.
The tether space elevator consists of a cable anchored to the Earth's surface, reaching into space. By attaching a counterweight at the end (or by further extending the cable for the same purpose), centrifugal force ensures that the cable remains stretched taut, countering the gravitational pull on the lower sections, thus allowing the elevator to remain in geostationary orbit. Once beyond the gravitational midpoint, carriage would be accelerated further by the planet's rotation.
The most common proposal is a tether, usually in the form of a cable or ribbon, spanning from the surface to a point beyond geosynchronous orbit. As the planet rotates, the inertia at the end of the tether counteracts the centripetal force of gravity and keeps the cable taut. Vehicles can then climb the tether and escape the planet's gravity without the use of rocket propulsion. Such a structure could theoretically permit delivery of cargo and people to orbit with transportation costs a fraction of those of more traditional methods of launching a payload into orbit. (Source: Wikipedia)