The concept behind the space elevator was first thought of by Russian scientist Konstantin Tsiolkovsky in 1865. He was fascinated by the Eiffel Tower and began to conceptualize a tower that would reach all the way into space up to a distance of 35,790km, he deemed this the “celestial castle” as he planned on having a castle at the end of the cable in perpetual orbit with the earth. According to his early plans the tower would be able to launch objects into orbit without a rocket. This is due to the proposition that the the elevator would achieve terminal escape velocity as the object ascended the cable, and would be released at the top and would remain in geostationary orbit because he suspected that the velocity to escape the atmosphere of the earth and the velocity necessary to enter geosyncronous orbit were the same. This concept varries greatly from todays concept of a space elevator.
As time went on the “Celestial Castle” concept was becoming less feasible by the day because there was no material in existence that had the necessary compressive strength to support its own weight let alone the weight of an object traveling along it. But in 1959 there was a tremendous breakthrough Yuri Artsutanov a space physicist from Russia came up with the idea to use a counterweight on earth to keep the cable motionless on earth.
It wasn't until 1969 at the NASA Ames Research Center that the concept of the Space elvevator as we know it today was developed by Jerome Pearson, President of STAR, Inc.. His idea somewhat of a "Stairway to Heaven" if you will was almost identical to Artsutanov's with the excetpion that the "cable" would be fixed to the earths surface with a counterweight and then would have another counterweight attached to the other end in space which would remain in geostationary orbit. Also his plan called for a cable with a varying width that would be widest at the point of orbit because that is where the highest amount of tension is. But in a sad reality none of these ideas had any true feasibility to them because just like Tsiolkovsky there was still no material capable of withstanding the necessary forces.
It wouldn't be until the invention of carbon nanotubes in the 1990's that the space elevator concept in general would hold any water. But since then theres been a virtual explosion on the space elevator market. Most concepts are based off of Pearson's plan using a cable formed from carbon nanotubes and an array of different types of "climbers" all of which will be further elaborated on in the Science section.