The henna plant produces a dye called lawsone that is found in the leaves of the plant. The leaves are harvested, dried, and ground into a fine powder. The powder is mixed with a liquid (water, lemon juice, tea, coffee ... there are as many recipes as there are henna artists!) to break down the plant cells and release the dye molecules. Many artists also add essential oils to their mix. Several types of essential oils contain something called monoterpene alcohol that, when mixed with the henna paste before dye release, actually improves the quality of the final stain.
The paste is then applied to the skin using various methods - anything from a small Mylar cone, to a "jack bottle" (something that looks like a Puffy-Paint bottle), to a stick. The paste should be left on the skin for at least several hours. During this time, the dye molecules in the paste bind to the keratin in the top layer of skin - the layer that exfoliates. After several hours, the paste can be brushed off. The resulting stain usually starts out as a light orange, and darkens to its final color over the course of a day or so.
The natural properties of each person's skin affects the result differently, but on the majority of people the final color of the stain will range between a warm, reddish chocolate brown and a deep, rich burgundy. The thicker the skin, the darker the stain is likely to be. On a rare few people, the stain will actually turn out a bright cherry red, but in over 10 years of working with henna, I have not yet seen this phenomenon in person.