A unicorn wand is any wand that has a small sliver of unicorn horn, one or more hairs from a unicorn's main or tail, or a sliver of unicorn hoof in its core.
Unicorns
Unicorn
A unicorn is a magical white horse with a single large, pointed, spiraling horn projecting from the center of its forehead and cloven hooves like a goat. Unlike the mares, unicorn stallions also have a goat's beard. Unicorns are symbols of purity due to the fact that their coats and horns are of the purest white.
Entering our world millennia ago through the Silver River portal in the Green Hills province in Faerie, unicorns first settled along the Garonne River near the present day city of Bordeaux France. From there, they spread eastward into the forests of Germany, Austria, Switzerland, and Hungary.