Plants have several methods for expanding and moving around. One way plants is expand is by extending stems out sideways from the plant. These take root and start a new plant at that point.

Rhizomes are modified stem tissue which moves out from the plant underground. Quackgrass moves around quickly by rhizomes. Some species and cultivars of hostas multiply by rhizomes and are, therefore, properly known as rhizomatous.

Many people mistakenly call these hostas stoloniferous. A stolon is a modified stem that moves above the ground. Examples would be goutweed (Aegopodium), Ajuga reptans, Lamium sp. and several other common groundcovers.

Note: I did a Google search on stoloniferous plants and it showed many, many incorrect listings that confused stoloniferous with rhizomatous.  There was even reference to something called a "stoloniferous rhizome". As I recall, I was always taught that it was either one or the other but I can't figure out a way that a plant part can be both above and below the ground at the same time. Oh, well. - Mr PGC

Other Hostas - Most hostas form a solid, thickish crown beneath the soil surface. This is a "swollen" structure that looks a little like a potato (but it is NOT a tuber) where the roots emerge from the bottom and buds expand from the top. These types of hosta expand by multiplying the number of buds which results in expansion of the crown from the center outward.