Scrub beets; trim stems to 1/2 inch. Place beets in a medium saucepan; add water to cover by 1 inch. Bring to a boil over high heat. Reduce heat to medium high; cook at a gentle boil until beets are tender when pierced with a small, sharp knife, 30 to 40 minutes. Remove from heat, drain, and set beets aside until cool enough to handle.
Meanwhile, grate horseradish on small holes of a box grater. Place in a small bowl; stir in vinegar, salt, and sugar.
Peel beets; grate on small holes of box grater. Add to horseradish mixture; stir well to combine. Keep covered to preserve flavor, and refrigerate until needed.
You can also make horseradish using other vegetables such as zucchini, carrots or whatever your imagination can come up with. I've made those 3 but always favor the beet horseradish. One of the other veggie flavored horseradishes is a nice dish to bring to someone's home. They get a big kick out of it.