Kutani ware has been painted in Ishikawa, Japan since the 1650s, and the incense burner (koro) is its jewel format - small enough to paint densely, important enough to deserve gold.
Five patterns cover most of what you will meet. Hanazume (flower-filled): blossoms packed edge to edge, often gold-outlined - the showstopper. Aochibu (blue dot): thousands of raised dots applied one by one, a test of nerve. Sometsuke landscape: underglaze blue mountains and water, the quiet one. Yoshidaya greens: the deep emerald palette. Shoza style: everything at once, red and gold and detail - maximal, historical.
A burner is judged by lid fit, foot finish, and how the painting wraps the curve. Use it or not; many collectors never burn a stick and simply live with the object.
Hand-painted Kutani burners and lucky cats from the Juttoku atelier are in Heritage Select; sticks to burn in them are in The Scent.