Disclaimer: I gathered the material from various sources from around the world, and pieced together a few myths.
How to describe a Unicorn
The word unicorn, in Latin, means 'one-horned'. It comes from the two words: unus, meaning one; and cornu, meaning horn.So far, a unicorn is a creature with one natural horn in the center of it's forehead. That's a good start, let's go deeper. A unicorn can be compared to other horned animals such as a ram or a rhino. A ram has two horns twisting around themselves ( not each other). A rhino has one horn at the end of its face. But, unlike the rhino, the unicorn has two horns; not a fact usually considered or known when thinking of a unicorn. The horns of a unicorn slowly grow into each other, twisting and raveling into one another to form to a point at adulthood. Thus, creating one horn. Unicorns have only been depicted around the northern hemisphere; China, Mongolia, the Middle East, North Africa, etc. So, they seem to only have migrated to an extent. Many of the writers of unicorns used colors besides the well known white, to describe a unicorn. In India it was said to have a white body, a red head, and sparkling blue eyes. But most of Europe seemed to agree to the pure white classification. In Europe the unicorns would live in temperate forests, and was said to be so fast that no hunter ( horse-back or not) could catch it.
Horn
In the early 16th Century the horn of the unicorn was prized and wanted. Poachers scouted forests, theives sold fakes. Elizabeth the first even had a 150,000 pound request for the horn of a unicorn, as it was said to hold magical powers. The horn of the Unicorn is white at birth. It stays white up until it's 100th birthday when it starts to turn from white to grey to black. It is written that a Unicorn with a pure black horn will turn into a Shadow; a huge, black, horse-creature with a long black horn and and eyes of gold. It is also written that it is given the power to control life and death.
Behavior
Unicorns will not accept contact with humans, except those of pure heart. They do, however, have contact with the elves and other elemental spirits.
Power
A unicorn is an untamed creature with strong elemental powers. It purifies through the powers of destruction, like phoenixes bring life with death. It tears down in order to reform.
Unicorns gaurd the way to the Faery realms and can guide you there if you are prepared for the journey.