Basically it is based on a flask with a tube that is able to refill itself again and again as seen below:
It was thought that the capillary action of the tube the water flowing in the tube, but since the same cohesion force that draws the liquid up the tube in the first place holds the droplet from releasing into the bowl, the flow is not perpetual. But who knows what kind of liquids they tried.
Interesting enough some people managed to make it recently and post it on youtube:
This video was made by Munchausen Today. They publish videos created on pseudo-scientific beliefs, unproven facts, false evidences, not-patented inventions and their own crazy ideas. They show that Boyle's thermodynamically impossible device is finally realized. With beer.
BUT WAIT. They cheated. There is a pump. This machine with beer would not really work.
The concept was that capillary action, which creates the meniscus of liquid seen in containers and is responsible for the flow of water from a tree's roots upward against gravity, would allow the thin side of the flask to draw fluid up and refill the cup side. In reality, this is not possible because surface tension will hold it in a droplet at the end of the tube rather than letting it fall. In the video above, the hydrostatic equation is used to suggest that the device works with carbonated beverages (it doesn't; the video's apparatus has a hidden pump) because the weight of the liquid is much greater than that of the foam. Of course, the hydrostatic equation doesn't apply to a flowing liquid!Incidentally, Boyle's Flask would work and run perpetually if it were loaded with a superfluid like liquid Helium, which under the right conditions (like 2 deg above Kelvin) it could have a viscoscity of zero. Liquid Helium can defy gravity (almost like capillary action) and move on the outside of the container as well as making a frictionless fountain. This would indeed give an endless supply of energy. The only problem: How much energy is it required to maintain the superfluid in a fluid state at 2 degrees above absolute 0?
Answer that and you will rule the world.
Read more at FYFD.