Hand boilers are used to demonstrate vapor-liquid equilibrium. The colored liquid in the boiler is comprised of a volatile mixture of liquids that have a boiling point close to room temperature.
When someone places their hands in the lower chamber, body heat increases the gas temperature and causes the gas to expand. The heat also causes some liquid to evaporate, which further expands the gas and pushes the liquid upwards.
When released, equilibrium is re-established. The gas condenses into a liquid again and flows back into the lower bulb.
The hand boiler can also be used to show properties of distillation by turning it upside down and placing the (now) bottom chamber in ice water.
The gas will condense in the cooled chamber. Since the dye evaporates at higher temperatures, the liquid that condenses in the cooled chamber is colorless.