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.
There are some differences between runners and rhizomes. Runners are a type of stolon that exist above ground and are produced by many plants, such as strawberries. Typically, stolons have very long internodes that form new plants at the ends. These rise to the soil surface and can produce foliage and flowers. In contrast, rhizomes most often have short internodes with leaf-scars and thin paper-like leaves and root along the under side of the stem.