Not to go too geek here, but the style of martial arts Mr. Miyagi taught is Goju-Ryu, an off-shoot of what is commonly referred to as Okinawan Kenpo. Okinawan Kenpo was "created" when Shaolin Monks traveled to Okinawa and taught the farmers their style of martial arts and adapted it to their every day tasks. ChÅjun Miyagi later modified it again and called it Goju-Ryu. The "Mr. Miyagi" character is based off of ChÅjun Miyagi.
Kenpo, or sometimes called Kempo, is the Japanese translation for the Chinese word Chuan'fa. It means "Chinese Hands"
Kenpo/Kempo. Shaolin Chuan'Fa, and the Korean art of Tang Soo Do (Way of Chinese Hands) are all basically the same Martial Art. The Japanese and Korean styles have been adapted for the region and lifestyles. Chuan'Fa later had its name changed to "Kung Fu" due to government fear of a rebellion. That's an even longer story than I've already typed out.