The definition of wave height is actually the vertical distance between trough and crest for wind waves and swells in units of meters or feet.
From a surfers perspective it is entirely from the individuals point of view as everyone has already stated. I find that the best way to state the size of the waves is to describe the lip of the breaking wave to how far up your body it is when you are standing at the bottom of the wave when you are upright. eg waist high, sholder high, double overhead etc.
The bottom line is that if you ask a surfer how big the surf is and they say 4 foot it could be anywhere between waist high and well overhead.
Brian Kennedy wrote:length of a wave
If you mean the length of the wave - eg distance a surfer can ride for (and not wavelength) then this is measured by the distance the wave breaks parallel to the beach until the wave is completely whitewater.