A space elevator is, in simplest terms, an elevator that rises above a planet's atmosphere and into space. It is also sometimes called a geosynchronous orbital tether or a Beanstalk (after the fairy tale Jack and the Beanstalk in which Jack climbs a magical beanstalk which has grown into the clouds). It is one of the varieties of a skyhook.
The concept of the space elevator first appeared in 1895 when a Russian scientist named Konstantin Tsiolkovsky was inspired by the Eiffel Tower in Paris to consider a tower that reached all the way into space. He imagined placing a "celestial castle" at the end of a spindle-shaped cable, with the "castle" orbiting Earth in a geosynchronous orbit (i.e. the castle would remain over the same spot on Earth's surface). The tower would be built from the ground up to an altitude of 35,800 kilometers (geostationary orbit). It would be similar to the fabled beanstalk in the children's story "Jack and the Beanstalk," except that on Tsiolkovsky's tower an electromagnetically driven elevator would ride up the cable to the "castle". Comments from Nikola Tesla are suggestive that he may have also conceived such a tower. His notes were sent behind the Iron Curtain after his death.
Tsiolkovsky's tower would be the able to launch objects into orbit without a rocket. Since the elevator would attain orbit velocity as it rode up the cable, an object released at the tower's top would also have the orbital velocity necessary to remain in geosynchronous orbit.
Building from the ground up, however, proved an impossible task; there was no material in existence anywhere with enough compressive strength to support its own weight under such conditions. It took until 1957 for another Russian scientist, Yuri N. Artsutanov[?], to conceive of a more feasible scheme for building a space tower. Artsutanov suggested using a geosynchronous satellite as the base from which to build the tower. By using a counterweight, a cable would be lowered from geosynchronous orbit to the surface of Earth while the counterweight was extended from the satellite away from Earth, keeping the center of mass of the cable motionless relative to Earth. Artsutanov published his idea in the sunday supplement of Komsomolskaya Pravda (Young Communist Pravda) in 1960.
Making a cable over 35,000 kilometers long is a difficult task. In 1966, four American engineers decided to determine what type of material would be required to build a space tower, assuming it would be a straight cable with no variations in its cross section. They found that the strength required would be twice that of any existing material including graphite, quartz and diamond.
In 1975 another American scientist, Jerome Pearson[?], designed a tapered cross section that would be better suited to building the tower. The completed cable would be thickest at its center of mass, where the tension was greatest, and would narrow to its thinnest at the tips to reduce the amount of weight that the middle would have to bear. He suggested using a counterweight that would be slowly extended out to 144,000 kilometers (almost half the distance to the Moon) as the lower section of the tower was built. Without a large counterweight, the upper portion of the tower would have to be longer than the lower due to the way gravitational and centrifugal forces change with distance from Earth. His analysis included disturbances such as the gravitation of the Moon, wind and moving payloads up and down the cable. The weight of the material needed to build the tower would have required thousands of Space Shuttle trips, although part of the material could be transported up the tower when a minimum strength strand reached the ground or be manufactured in space from asteroidal or lunar ore.
Arthur C. Clarke introduced the concept of a space elevator to a broader audience in his 1978 novel, The Fountains of Paradise, in which engineers construct a space elevator on top of a mountain peak in the equatorial island of Taprobane (closely based on Sri Lanka).
David Smitherman[?] of NASA/Marshall's Advanced Projects Office has compiled plans for such an elevator that could turn science fiction into reality. His publication, "Space Elevators: An Advanced Earth-Space Infrastructure for the New Millennium"  (http://flightprojects.msfc.nasa.gov/fd02_elev), is based on findings from a space infrastructure conference held at the Marshall Space Flight Center in 1999.
Yet another American scientist, Bradley Edwards[?], suggests creating a 100,000 km long paperthin ribbon, which would stand a bigger chance of surviving impacts by meteors. The work of Edwards through Eureka Scientific  (http://eurekasci.com/SPACE_ELEVATOR/Space_Elevator_Homepage) and HighLift Systems  (http://www.highliftsystems.com) has expanded to cover: the deployment scenario, climber design, power delivery system, orbital debris avoidance, anchor system, surviving atomic oxygen, avoiding lightning and hurricanes by locating the anchor in the western equatorial pacific, construction costs, construction schedule and environmental hazards. Plans are currently being made to complete engineering developments, material development and begin construction of the first elevator. Funding to date has been through a grant from NASA Institute for Advanced Concepts. Future funding is sought through NASA, DoD, private and public sources.
A space elevator could also be constructed on the other planets, asteroids and moons.
A Martian tether could be much shorter than one on Earth. Mars' gravity is 30% (approximately 1/3) of Earth's, while it rotates around its axis in about the same time as Earth. Because of this, Martian geostationary orbit is much closer to the surface, and the elevator would be much shorter.
A Lunar elevator would not be so lucky. Since the Moon's rotation keeps the same face towards the Earth, the center of gravity of the elevator would need to be at the L1 or L2 Lagrangian points, which are special stable points that exist about any two orbiting bodies where the gravitational and rotational forces are balanced. The cable would point either to Earth for the L1 point, or face away from Earth for the L2 point. However, due to the lower gravity of the Moon, the total mass of a Lunar cable could be dramatically less than the mass of an Earth-based elevator, since less material would be needed in order to provide the necessary tensile strength to support itself against lunar gravity. Without a counterweight the 'L1'-cable would have to be 291,901 kilometers long and the 'L2'-cable would have to be 525,724 kilometers long. Considering that the distance between the Earth and the Moon is 351,000 kilometers, that's a long cable. Far shorter cables, perhaps not more than twice the length of the ~60,000 km distance to the L1 or L2 points of the Earth-Moon system would suffice if a large counterweight of lunar-derived materials were placed at the end of the cable.
Rapidly spinning asteroids or moons could use cables to eject materials in order to move the materials to convenient points, such as Earth orbits; or conversely, to eject materials in order to send the bulk of the mass of the asteroid or moon to Earth orbit or a Lagrange point. This was suggested by Russell Johnston in the 1980s. Freeman Dyson has suggested using such smaller systems as power generators at points distant from the Sun where solar power is uneconomical.
We can determine the orbital velocities that might be attained at the end of Pearson's 144,000 km tower (or cable). At the end of the tower, the tangential velocity is 10.93 kilometers per second which is more than enough to escape Earth's gravitational field and send probes as far out as Saturn. If an object were allowed to slide freely along the upper part of the tower a velocity high enough to escape the solar system entirely would be attained. For higher velocities, the cargo can be electromagnetically accelerated, or the cable could be extended, although that might necessitate counterweights below geosynchronus orbit in order to maintain the structure's center of gravity at geosynchronus orbit, and would require additional strength in the cable.
NASA has identified "Five Key Technologies for Future Space Elevator Development":
Carbon nanotubes have exceeded all other materials and appear to have a theoretical strength far above the desired range for space elevator structures, but the technology to manufacture bulk quantities and fabricate them into a cable has not yet been developed.
With Space Elevators like this, Humans can send materials into orbit at a fraction of the current cost (from around $30000 today to $3 per kg, a factor of 104!); the marginal cost of a trip would consist solely of the electricity required to lift the elevator payload, some of which could be recovered by using descending elevators to generate electricity as they brake, or generated by masses braking as they travel outward from geosynchronous orbit (A suggestion by Freeman Dyson in a private communication to Russell Johnston in the 1980s.) This means that hospitals, mining facilities, international trade, and travel could all be done in space with the help of these space elevators.
As with any structure there are a number of ways in which things could go wrong. A space elevator would present a considerable navigational hazard, both to aircraft and spacecraft. Aircraft could be dealt with by means of simple air-traffic control restrictions, but spacecraft are a more difficult problem. Over a long period of time all satellites with perigees below geostationary altitude would eventually collide with the space elevator, as their orbits precess around Earth. Most active satellites are capable of some degree of orbital maneuvering and could avoid these collisions, but inactive satellites and other orbiting debris would need to be either preemtively removed from orbit by "garbage collectors" or would need to be closely watched and nudged whenever their orbit approches the elevator. The impulses required would be small, and need be applied only very infrequently; a laser system may be sufficient to this task.
Meteoroids present a more difficult problem, since they would not be predictable and much less time would be available to detect and track them approaching Earth. It is likely that a space elevator would still endure impacts of some kind, no matter how carefully it is guarded. However, most space elevator designs call for the use of multiple parallel cables separated from each other by struts, with sufficient margin of safety that severing just one or two strands still allows the surviving strands to hold the elevator's entire weight while repairs are performed. If the strands are properly arranged, no single impact would be able to sever enough of them to overwhelm the surviving strands.
A further potential risk of structural failure comes from the possibility of vibrational harmonics within the cable. Like the shorter and more familiar strings of stringed instruments, the cable of a space elevator has a natural resonance frequency. If the cable is excited at this frequency, for example by the travel of elevators up and down it, the vibrational energy could build up to dangerous levels and exceed the cable's tensile strength. This can be avoided by the use of intelligent damping systems within the cable, and by scheduling travel up and down the cable keeping its resonant frequency in mind.
If despite all these precautions the elevator is severed anyway, the resulting scenario depends on where exactly the break occurred. If the elevator is cut at its anchor point on Earth's surface, the outward force exerted by the counterweight would cause the entire elevator to rise upward into a stable orbit. The ultimate altitude of the lower end of the cable would depend on the details of the elevator's mass distribution. In theory, the loose end might be secured and fastened down again. This would be an extremely tricky operation, however, requiring careful adjustment of the cable's center of mass to bring the cable back down to the surface again at just the right location. It may prove to be easier to build a new system in such a situation.
If the break occurred at any altitude up to about 25000 km, the lower portion of the elevator would descend to Earth and drape itself along the equator while the now unbalanced upper portion would rise to a higher orbit. Some authors have suggested that such a failure would be catastrophic, with the thousands of kilometers of falling cable creating a swath of meteoric destruction along Earth's surface, but such damage is not likely considering the relatively low density the cable as a whole would have. The risk can be further reduced by triggering some sort of destruct mechanism in the falling cable, breaking it into smaller pieces.
Any elevator pods on the falling section would also reenter Earth's atmosphere, but it is likely that the elevator pods will already have been designed to withstand such an event as an emergency measure anyway. It is almost inevitable that some objects - elevator pods, structural members, repair crews, etc. - will accidentally fall off the elevator at some point. Their subsequent fate would depend upon their initial altitude. Except at geosynchronous altitutde, an object on a space elevator is not in a stable orbit and so its trajectory will not remain parallel to it. The object will instead enter an elliptical orbit, the characteristics of which depend on where the object was on the elevator when it was released.
If the initial height of the object falling off of the elevator is less than 23000 km, its orbit will have an apogee at the altitude where it was released from the elevator and a perigee within Earth's atmosphere - it will intersect the atmosphere within a few hours or even minutes, and not complete an entire orbit. Above this critical altitude, the perigee is above the atmosphere and the object will be able to complete a full orbit to return to the altitude it started from. By then the elevator would be somewhere else, but a spacecraft could be dispatched to retrieve the object or otherwise remove it. The lower the altitude at which the object falls off, the greater the eccentricity of its orbit.
If the object falls off at the geostationary altitude itself, it will remain nearly motionless relative to the elevator just as in conventional orbital flight. At higher altitudes the object would again wind up in an elliptical orbit, this time with a perigee at the altitude the object was released from and an apogee somewhere higher than that. The eccentricity of the orbit would increase with the altitude from which the object is released.
Above 47000 km, however, an object that falls off of the elevator would have a velocity greater than the local escape velocity of Earth. The object would head out into interplanetary space, and if there were any people present on board it may prove impossible to rescue them.
All of these altitudes are given for an Earth-based space elevator, a space elevator serving a different planet or moon would have different critical altitudes where each of these scenarios would occur.
Another type of space elevator that doesn't rely on materials with high tensile strength for support is the space fountain[?], a tower supported by interacting with a high-velocity stream of magnetic particles accelerated up and down through it by mass drivers. Since a space fountain is not in orbit, unlike a space elevator, it can be of any height and placed at any lattitude. Also unlike space elevators, space fountains require a continuous supply of power to remain aloft.
Still smaller-scale tether propulsion is a possible propulsion method for spacecraft in planetary orbit.