I've seen an hypothetical "Space elevator" referenced in various media the last couple of years. Is this something thats really planned in the making, or just a hypothetical scenario? How will a space elevator work, if it's even theoretically possible to make one?
The idea behind a Space Elevator is to build a structure, usually from the surface of the Earth (Though there are notions of ones that just touch the lower atmosphere) to Geosynchronous orbit (22,000Miles) and then to balance the weight out, a similar distance of structure continuing on a further 22,000 miles (or else a large counterweight)).
It needs to be GeoSync, so that it stays in orbit over the same spot on Earth, if you wish to attach it to the ground.
Now attach a platform that is powered and can climb up and down this structure and you have a functional elevator to GeoSync orbit.
In principal solar or beamed laser power is all you need to power the climb and descending. Consider speed for a moment? How fast can you build a climbing device? It has to go 22,200 Miles. That can take a long time, particularly if you are touching something (the elevator structure itself) and are friction limited.
What should it be made of? Well right now we do not have anything strong enough, that could be manufactured (I note that some carbon nanotube structures have the appropriate strength, but we cannot manufacture large enough structures of them (yet)) to carry the needed load.
As Sparr answered, the moon or Mars are easier targets due to lower LunaSync or AreoSync orbits and lower gravity, thus lowering the structural strength required.
Lots and lots of great Sci Fi stories about it.
Kim Stanley Robinson in his "Mars" series has one at Mars that eventually is destroyed by terrorists, and it wraps around the planet several times in the process. (Consider, that the circumference of the Earth is about 24,000 miles, which is pretty darn close to the height of GeoSync. So almost twice around the Earth (Remember the extension out the back side for balance))
A space elevator is a hypothetical mode of transportation between Earth's surface and space. While the idea was popularized by science fiction author Arthur C. Clarke, the idea is much older.
Basically, a space elevator is exactly what it sounds like. An elevator that takes you from the Earth surface up to a geostationary satellite or space station. In theory, it would make lifting payloads up to orbit cheaper than using rockets as we do today.
One way to do this is to capture an asteroid and place it in a geostationary orbit above the tropic. A cable is then constructed stretching from a station on the asteroid to another station down on Earth. The asteroid thus becomes a counter weight for the cable. Imagine twirling a yo-yo around yourself. Transporting things between the two stations could be done via using climber vehicles that climb the cable.
As you can imagine there are significant engineering hurdles that need to be tackled before something like a space elevator can be constructed. One significant hurdle is the cable itself, or rather the material used in constructing it. The only natural substance strong enough to be used for the space elevator is diamond. Of synthetic materials carbon nanotubes are a possible candidate, though we still do not have a way of constructing long lengths of it yet.
As of now I know of only two groups attempting a space elevator: The LiftPort Group are focusing on carbon nanotube research and are contemplating a lunar elevator as a preview for a terrestrial one, and the Obayashi Corporation which has set a goal date of 2050 for it's space elevator.
There are plenty of other good questions (and answers) here in Space.SE about space elevators, so you may want to read those as well.
A space elevator on Earth is, for now, science fiction. We have the technology to build one on the Moon, though. Building one on any body would greatly reduce costs to launch payloads into orbit or beyond, so this is definitely a goal and not just a hypothetical.
As to how it would work, imagine a cross between an elevator and a train. Most of the elevator would be outside of the atmosphere (if the body has one at all), so the cars could move very quickly without encountering air resistance.
A space elevator is an tether with climber that connects a planet or other astronomical body (such as the moon) past a balance point (such as a Lagrange point or geosynchronous orbit) to allow for "landing" and "takeoff" of payloads.
Although an Earth elevator is largely hypothetical at this point (being dependent on carbon nanotubes, boron nanotubes, or graphene being able to be manufactured to appropriate lengths), a lunar elevator doesn't have as extreme of a material requirement, and vectran string is strong enough to meet its requirements, and so a lunar elevator's issues aren't hypothetical problems so much as practical problems (where to get the funding to build it, specifically.)
The Lunar elevator has the distinct advantage, also, that it may turn the Earth Elevator from a hypothetical problem into a practical problem as well. The reason being is that a waystation located at the L1 Lagrange point of the Lunar elevator would be able to sustain a pico-gravity facility, which would make precision creation of CNTs much more feasible, allowing creation of the necessary parts of for an Earth Space Elevator.
And space elevators are actually planned. Japan is planning an Earth elevator by 2050, and LiftPort in the U.S. is aiming for a Lunar one by 2020 (although funding problems may prevent the latter.)
There have also been hints that Russia, EU, and China may pursuing space elevator concepts (the EU poured a lot of money into graphene research referencing both computing and aerospace tech) and Russia has been investing a lot into space development recently, supposedly looking at colonization, and this started shortly after papers on the feasibility of a Lunar Elevator were publicized. China has also declared an interest in setting foot on the Moon, and due to the high expense of shooting up enough fuel for retrograde burn and return, a Lunar elevator would be the cheaper route to pursue that goal. Although it has yet to be formally announced in the case of all three, speculation is that they will turn to a lunar elevator concept due to its feasibility and lower cost and benefit to their other goals.