There are many variations of surfing, and the definition for what constitutes a suitable wave and craft are purely subjective. In other words, the term "surfing" refers to the act of riding a wave and not the form (with or without a board) in which the wave is ridden.
For instance, the native peoples of the Pacific surfed waves on alaia, paipo, and other such crafts on their belly, knees, and feet. Not to mention, Bodysurfing, the act of surfing a wave without a board, is considered by some to be the purest form of surfing. That much said, the more modern day definition of surfing tends to refer to when a surfer rides a wave standing up on a surfboard, which is referred to as stand-up surfing.
Although, another prominent form of surfing in the ocean today includes bodyboarding, which refers to when a surfer rides a wave either on the belly, dropknee, or stand-up on a bodyboard. Not to mention, knee boarding, surfmatting (riding inflatable mats), foils, and so forth.
Three major subdivisions within stand-up surfing are longboarding, shortboarding, and stand up paddle surfing (SUP), reflecting differences in board design, including surfboard length, riding style, and the kind of wave that is ridden.
Source: Wikipedia, Facebook