The idea that the Earth is hollow is a persistent one which pops up occasionally in cultures throughout history. The belief itself is simple: that our planet is not a solid sphere filled with molten rock, but is in fact more like a tennis ball: a hollow sphere with lots of space inside.
In most telling, the center of the Earth contains another sun which floats in the middle, bringing light and warmth to the inner Earth. (This sun is a key element, obviously, because otherwise how would all the plants grow? DUH.)
Needless to say, a basic understanding of physics and gravity is enough to dispel any lingering thoughts that our planet might be hollow. But many people don't let the facts get in the way of their beliefs.
Hitler may have believed in a hollow Earth. (Let's face it - it's less crazy than some of the other stuff he believed.) Nazi Admiral Donitz once gave a speech which seemed to indicate that Hitler was sending submarines out to the seas beneath the polar ice caps to look for the entrance.
The hollow Earth naturally must be accessible somehow, or else where's the fun? Most people seem to believe that there are openings at the poles of various sizes. In the days before satellite imagery, a lot of people believed that the holes were very large - thousands of feet across. One noted hollow Earth scholar proposed that many Ice Age species didn't go extinct - they just wandered into the holes and never came out.
These days most people believe that they are small, or hidden somehow, whether beneath the ice or in another dimension. Others, that entrance can be gained by climbing down through a dormant volcano, or by becoming an ascended master able to travel through the various trans-dminensional planes.
There is a long-standing theory among certain UFO camps that UFOs do not in fact come from outer space, but are instead being built and deployed from within the hollow Earth. They cannot be said to be piloted by aliens, then, but they are almost certainly not being piloted by other humans. (But surely a parallel civilization which evolved inside our planet could be considered "alien"?)
These beliefs often cite Agartha, the dominant society within the hollow Earth. Its capital city is Shambhalla, and its residents may be a superior race which arose on Earth millions of years ago but took refuge inside our planet for various reasons. Some say they were driven underground by ancient wars (such as the war between Atlantis and Lemuria). Others believe that they are hiding from a hostile force, or simply that they found our sun's rays damaging.
One notable branch of hollow Earth theory states that Mount Shasta in California is itself hollow, and is in fact a giant mountain-shaped civilization all on its own. Inside Mount Shasta you can allegedly find the Agarthean city of Telos, which believers have described and fleshed out in a surprisingly intricate fashion over the years.
Illustration credit: Flickr/kodunion