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The effect relies on corona discharge, which allows atoms to become ionized very easily near sharp points and edges – this is why lightning rods are pointy. In this case two electrodes are used with a high voltage between them, about 20kV, where one electrode is small or sharp, and the other larger and smoother. This creates a high field gradient around the smaller, positively charged electrode. Around this electrode, electrons are stripped off the atoms in the surrounding medium, they are literally pulled right off by the electrode's charge. The electrons quickly move to the electrode, and are driven to the negative electrode by the voltage.
This leaves a cloud of positively charged ions in the medium, which are attracted to the negative electrode. This also drags along some of the surrounding medium, causing what is known as ion wind, which creates a breeze of considerably greater magnitude than the ions themselves account for. This effect can be used for propulsion (see Lifter) and fluid pumps.
The effect has become something of a cause celebre in the UFO world, where it is seen as an example of something much more exotic than electrokinetics. Charles Berlitz devoted an entire chapter of his book The Philadelphia Experiment to a retelling of Brown's early work with the effect, implying he had discovered some new electrogravity effect being used by UFOs. In fact Brown was fully aware of how the device worked, but that makes for a less interesting story. Today the internet is filled with sites devoted to this interpretation of the effect (see below for some of the more profound sites).
Some people think that the Tesla coil might be related to this effect. In fact, when Tesla came to the USA he was supposidly carring plans for a "flying machine". But the only thing the Tesla coil and the Biefeld-Brown effect have got in common is that in both of them, alternating high voltage plays a vital role.
T. T. Brown issued a number of patents on his discovery: