The folks at ISR or Institute for Scientific Research explain the concept behind the Space Elevator in this Video.
The space elevator in the simplest terms is a Ribbon with one of its ends attached to the Earth and the other end attached to a counterweight in Space. The rotation of the Earth throws the counterweight on the Ribbon outwords, keeping the Ribbon taut.
This Ribbon is to be made of a carbon composite which is supposedly almost 30 times stronger than steel. The Ribbon would be 3 feet wide, thinner than a sheet of paper but almost 62000 feet long and able to take upto 20 tons of objects to space.
Using this concept, Mankind can persue a dream by climbing a Ribbon to the Stars
Watch the full video below