Root hormones stimulate the development of new roots from a cutting, allowing a part of a plant to grow a genetic copy of the original. The key ingredient in most rooting stimulants is a growth regulating plant hormone called auxin, which sends a signal to the plant's cells to stop differentiating into various elements (like leaves and branches) and start focusing energy on roots alone.
The best place to apply a rooting hormone is, unsurprisingly, the end of the plant that could most easily be planted. If you have a twig from a tree, for example, you can shave off some bark near the break point and apply root hormone to the exposed end. These substances are most helpful when applied to a fresh cut spot, not one that's been scabbed over with new growth.
If you use a powdered version, wet the end of the plant to encourage the hormone to stick. If you use a liquid version, dip the plant briefly—don't use it to stir the substance, and don't let it steep too long. It's possible to overdose a plant on auxins, killing part of it and maybe negatively affecting budding later on.
While it's not a fail safe solution, root hormone application increases the chance a clipping will grow new roots. It also appears to help plants form phloem and xylem (phloem helps move the sugars made from photosynthesis to the rest of the plant, and xylem helps move nutrients from the earth into the roots).
Indolebutyric acid and naphthaleneacetic acid are the two most common auxins on the commercial market. They're especially useful for stimulating rooting in plants that won't otherwise tend to do so. Most succulent plants, for example, will grow new roots from clippings in the right conditions. Many trees, however, are unlikely to grow roots without some coaxing. Some, like pine and fir trees, are very difficult to root from clippings, even with the help of growth hormones. Others, like willow and yew, might be very receptive to this kind of boost.
Willow trees, in fact, are so good at making auxins that some gardeners use cuttings from them to make natural rooting solutions for other plants.