Stolon is a term used often in botany. It describes a horizontal stem running across or just under the ground. A runner is a stolon which runs on the ground; a rhizome is the same thing, but running underground.
Plants with stolons or stolon-like rhizomes are called stoloniferous. A stolon is a plant propagation strategy. At nodes, rootlets are put down and new stems are put up. All these plant forms are clones from a single genetic individual.