The best way to process horseradish in a home kitchen is to grate it, either in a food processor, a blender or by hand with a metal grater, and soak it with vinegar and add salt. It is better to avoid overprocessing so the root doesn't become a pasty mush.
When you grate horseradish, you release the chemical responsible for its spicy pungency. As time passes, the chemical reaction that started during the grating process intensifies the heat from both its flavor and aroma, but adding vinegar stops the reaction. If you allow the root to sit longer after you grate it and before you add the vinegar, it will be stronger.