The term "living fence" can refer to a variety of different types of plantings. Some gardeners refer to any hedge as a "living fence". Numerous shrubs, e. g. privet, yew, box, arborvitae, beach, hornbeam, etc  can be pruned to form a hedge.

Others experts use the term to refer to a fence on which plants are growing.  A good example of the latter would be an espaliered small tree, such as a fruit tree, grown on a trellis. In an espalier, the tree or shrub is pruned and trained to grow on a support in a defined form. This website gives details on how to grow an espaliered plant.

An attractive living fence can easily be grown with willows (Salix spp.). There is even no need to buy rooted willow plants as willow canes inserted into the ground will root and grow rapidly. This website gives detailed instructions on how to do this.

Courtesy of NYBG Plant Information