Stolons are shoots that bend to the ground or that grow horizontally above the ground 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. "Rhizomes" means something similar, but in contrast to stolons, rhizomes dwell completely underground.
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's 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 don't want it! One way to stop the encroachment is to install lawn edging.
Likewise, when you wish to transform a section of your lawn into, say, a perennial bed, you must be ruthless in robbing the grass stolons of viability. I relate several methods in my full article on how to get rid of grass.