The common name, Witch Hazel, derives from the passing resemblance of the leaves to our common Hazel - though they are entirely unrelated - which led early settlers to America to use the shoots for water divining - this possibly earning it the 'witch' in its name. Another possible origin is the Middle English 'wiche' meaning pliant - with Witch Hazel thought to have been used as a synonym for Wych Elm that the American settlers simply transferred to the new shrub. But you may associate the name more with a variety of commercial medical products. It is mostly used as a lotion for applying to bumps and bruises - it's certainly in our medicine cupboard and has been splashed over the kids many a time, though I have no idea whether it actually does any good. It is also often used as a natural remedy for eczema, aftershave applications, ingrown nails, to prevent sweating of the face, cracked or blistered skin, and for treating insect bites.
The Witch Hazels are in the Hamamelidaceae family and are closely related to Persian Ironwood, Sweet Gum and Katsura - all excellent trees. There are four or five species in America, China and Japan and all are distinguished by their thin spidery petals. Some of the best known (and most easily available here) are cultivars of the hybrid Hamamemis x intermedia, which is a cross between the Chinese Hamamelis mollis and the Japanese Hamamelis japonica. To confuse matters, they can be grafted onto the American Hamamelis virginiana.
The flowers are also said to be beautifully scented, but for the life of me I can't smell a thing. Christopher Lloyd, in his 'The Well Tempered Garden' writes of bringing a branch into the house where the warmth releases the scent better - but he warns that the flowers will only last a fortnight indoors after which he is sorely tempted to harvest another, and then again and again until he has decimated the tree.