Here is how I envision a space elevator.
You have a base on the ground (or, more feasibly, the ocean). From that base, you have rails extending up into space. The rails would connect onto an orbiting station/satellite in space, that matches the orbit and trajectory of Earth.
Then, there would be a shuttle affixed to the railways. From the base, an explosion would propel it up, and then it would drop back down to transport people downwards.
Theoretically it is possible, but the funds and expenses are so great and it is not generally deemed as profitable enough to warrant enough support for a project like that.