It's a simple concept, really. Just follow this quick and dirty guide.

-You MUST be able to ride a bike with your hands.
I cannot stress this enough. I shudder while thinking of all the poor people who tried to ride their bikes without hands without knowing how to ride it with.
-You must have a fairly old bike.
The reasons for this are twofold. The obvious reason is that falling, something which you are bound to do, will not have as much monetary impact. The second reason is surprising -- it is much easier to ride an old bike without hands than a new one. For some reason the handlebars are stiffer and do not turn as much.
-You must not be afraid to fall.
The path to success is fraught with hardships, young grasshopper.

How to do it
It's rather simple, really. However, there are some things you must keep in mind. For one, get up to a good pace before removing your hands. If you find it hard to keep your balance, the most probable cause is that you simply are not going fast enough. Remember when you first learned to bike? You had to bike quickly to keep your balance. There is some scientific explanation for this.
Secondly, you must conciously keep your balance. For example, if you feel your bike tipping to the left, you must conciously change your centre of mass to the right. You do this unconciously while riding with hands, but you must learn it again from the start now.
And lastly, make sure you are in an open area without pedestrians. I'm sure you can figure this one out for yourself.

Biking without hands is rewarding and useful. For example, now I can drink caffeinated water and read my Calculus textbook at the same time, while biking. My next project is to figure out how to do this safely. However, do not let my isolated incident dissuade you. Go forth and bike!