It wouldn't be until the invention of the Carbon Nanotube that the Space Elevator would even have hope of becoming a reality, but nowadays it has progressed to the point that it looks like the next great inovation of space travel will be the Space Elevator especially with the impending retirement of the shuttle program with no feasible replacement yet in production. Currently the fore runner in this race is the LiftPort Group who plans to have a fully functioning elevator similar to the one below buy October 27, 2031.
In this basic elevator design there are a few yet all extremely technical compontents
1) Cable: in all this is probably the simplest component hear you would think, it's just a cable right? WRONG, this cable would be using the most advanced technology to date: nanotechnology. As seen on the history page nanotubes are constructed from carbon atoms at the nano level, most have a diameter of one nanometer with a cost of an astronomical $1500US per gram where as the average steel price is about $700US per meteric tonne, but there is justification for this cost because nanotube have a specific strength up to 48,000 kN·m·kg^-1 per gram versues steel which is right about 154 kN·m·kg^-1. This technological breakthrough is what gives the elevator feasibility because it allows the cable to withstand the necessisary forces without collapsing under it's own weight like it would if constructed from steel.
2) Climber: because a space elevator cannot opperate like a typical elevator in the sense of having a cable pull the car up the shaft it needs to have devices to climb the cable hence the "Climbers" for the most part the elevator is designed for the climber to only go upwards because the payload usually will not be coming back down. The proposed idea for powering the climbers also varies with the most accpeted being the climbers are wirelessly transfered electricity from the cable, although some argue that this is impractical and that the climbers could just be powered by large battery packs.
3)Counterweight:The Counterweight is the simplest part of the space elevator, so long as its an object of great mass and located in orbit of the Earth around the equator it will work, most proposed elevators use a harnessed asteroid or meteor as their counterweight. Although there are some that are proposing attaching buildings to the counterweight like cities in the sky similar to that of Cloud City in "Star Wars Episode V: The Empire Strikes Back".
For a more detailed explanation on the inner-workings of the Space Elevator please visit How Stuff Works