Water boils at 212F (at sea level). When water (or any liquid) is at a proper boil, the surface bubbles and the liquid beneath it churns vigorously.
Simmering is gentler. Water simmers at 180-190F, at which point a bubble make break the surface now and then (most likely at the edges of the pan) and the surface will shimmer, as though it is about to move.
Boiling is generally used to cook pasta and other grains, or to reduce and intensify a sauce. Maintaining a simmer can require close attention, because as heat builds in a pot, that simmer easily can turn to a boil. Simmering is often used for cooking proteins, which would toughen if subjected to the rigors of boiling.