A basket hilt is a type of hand protection for swords and maces in combat. They are rigid and often made welded bars, or a complete steel covering. Coincidentally, they look somewhat like a basket. The pros for basket hilts is that they give your sword weight at the bottom, and are very effective in protecting your hand from injury. The disadvantages of using a basket hilt is that they can be confining, weigh your sword down, and some find it restricts wrist movement.
Other types of hand protection which can be used are ice hockey gloves (aka 'gumby'), gauntlets or clam shells. A quillion can be used instead of a basket hilt so long as any of the above hand protections mentioned above are used.
Consult your local kingdom's marshal handbook for more information.