In The Hitchhiker's Guide to the Galaxy, Douglas Adams wrote that the trick to flying is to "throw yourself at the ground and miss." It turns out that from a physics viewpoint, this is very nearly correct.
The trick is to actually throw yourself really fast *sideways* such that when you fall, you miss, and keep falling, and keep missing.
Newton drew a picture of this in Principia. Imagine you fire a cannonball so fast that by the time it falls to the ground, it has gone so far that the curvature of the Earth means that the ground is farther away. 
If you could fire a cannonball fast enough from the top of an extremely high mountain (and air resistance didn't matter), it would go so far that it would miss the ground entirely, the pull toward the center of the Earth just constantly pulling it around, but never to the ground.
Astronauts in orbit around the Earth are not weightless. The gravity there is almost as strong as it is on the surface -- its keeping the space station in orbit. But the astronauts FEEL weightless, because they are falling. Just constantly falling, and missing the ground.
