Horseradish can also be prepared, more quickly and with much less pain, by using a food processor. After peeling and cleaning the root, as described in sneff's writeup above, chop the horseradish into chunks about an inch across, and toss into the food processor. Put in a teaspoon to a tablespoon of vinegar. A few quick pulses will reduce the chunks to a grated consistency. This is far less time-consuming and tear-generating than grating the root by hand.
Good, fresh horseradish is an essential ingredient for the best shrimp cocktail.