The Rubik's Cube is indeed solveable by normal people. In fact, some not-so-normal people (like me) enjoy speedcubing, or solving the cube as fast as possible. If you are interested in learning to just solve the cube, I recommend this beginner's solution. If you want to know more about speedcubing you should check out Speedcubing.com and its rather large links page.
I'm no longer as fast as I used to be due to lack of practice, though I can still whip out sub-20 second averages.
Best average of 12: 15.86 seconds
Best non-lucky single: 12.18 seconds
Best blindfolded: 2 minutes 56.86 seconds.
A selection of videos of myself solving cubes..
A list of algorithms I've memorized for the last layer.
An intermediate last layer method for those who want to go a little faster.
Solve a Rubik's Cube with a blindfold on