Stolons are shoots that bend to the ground or that grow horizontally above the ground (or slightly below it) and produce roots and shoots at the nodes. The term is most often used in discussing the botany of lawn grasses (although it can apply to other kinds of plants, as well) to describe a means by which they spread. A plant that spreads in this manner is said to be "stoloniferous" (adjective).
"Rhizomes" means something similar, but, in contrast to stolons, rhizomes dwell completely underground. Many of the worst invasive species have achieved that status thanks to their vigorous rhizomes; a notorious example is Japanese knotweed. The most aggressive types of grass, such as Bermudagrass, spread by both rhizomes and stolons.
Other kinds of plants that spread via stolons (also commonly called "runners") include:
Once you learn how greedily the stolons of bugleweed reach out to colonize new land, it becomes apparent why this flowering ground cover, so invasive in North America, makes the list of worst plants to grow in your yard. If you hike in the woods in eastern North America, you may have encountered the colonies that trout lily is so successful at forming -- for the same reason. As a native, we praise the colonizing habits of trout lily, while cringing at the thuggish behavior of the alien, bugleweed (it hails from the Mediterranean region). The principle behind the spreading is the same; how you judge the results (good or bad) is a matter of perspective.
Grass Stolons: a Good Thing, a Bad Thing
Spreading grass, too can be desirable or undesirable. It really depends on the context. First the good news: the ability of grass to spread in this way makes it possible for you to have a sensational lawn, a green carpet that fills in an area and has no bare patches. It is precisely grass' tendency to spread in this way that makes it such a widely used "floor" for outdoor living spaces. The best lawns are spaces where you can walk around barefoot and never even get dirt on your feet. You can thank the grass' stolons for that.
Now the bad news: grass stolons are not content merely to spread where you want them to. Rather, they will spread into garden areas adjacent to the lawn, as well. Since grass is unwanted in such spots (indeed, it is essentially considered a weed when growing in a vegetable patch, flower border, etc.), you have to take measures to control its penetration into those parts of your yard.
The fact that lawn grasses can spread via stolons causes a lot of extra landscaping work for gardeners. When opening up flower beds adjacent to lawn areas, it is important to arrest stolons in their tracks before they can creep into your flower beds. Once they break into your beds, they will become established and cause grass to appear -- precisely where you do not want it. One way to stop the encroachment is to install lawn edging. Whatever material that you use for edging, it needs to protrude above the surface in order to keep grass stolons from spreading into planting beds.
Likewise, when you wish to transform a section of your lawn into, say, a perennial bed, it is not enough simply to remove the green part of the lawn that is most visible: you must also be ruthless in robbing the grass stolons of viability. There are several methods for doing so. Want to learn about them? Read the full article on how to get rid of grass.