It is named after Johann Gottlieb Leidenfrost, who discussed it in A Tract About Some Qualities of Common Water in 1756.
To demonstrate the Leidenfrost effect at home, just take a normal clean frying pan and heat over a gas stove. Have a bowl of clean water handy. Every now and then, dip your fingers into the bowl of water and sprinkle a few drops onto the pan. Initially, as the temperature of the pan is below 100 °C, the water just flattens out and slowly boils away. As the temperature of the pan goes above 100 °C, the water drops hiss on touching the pan and evaporate relatively quickly. Later, as the temperature goes past 250 °C, the Leidenfrost effect comes into play. On contact the droplets of water do not evaporate away so quickly. This time, they bunch up into small balls of water and skitter around, lasting much longer than when the temperature of the pan was much lower. This effect lasts pretty much until a much higher temperature causes any further drops of water to evaporate too quickly to cause this effect.
This works because, at temperatures above the Leidenfrost point (about 220 °C for water), when water touches the hot plate, the bottom part of the water vapourizes immediately on contact. The resulting gas actually suspends the rest of the water droplet just above it, preventing any further direct contact between the liquid water and the hot plate and dramatically slowing down further heat transfer between them. This also results in the drop being able to skit around the pan on the layer of gas just under it.