Sorry to contradict, Alcatraz, but Shotokan was created by Gichin Funakoshi -- an Okinawan.
To be fundamentalist about it, the only true "Japanese" martial arts are Aikido, Judo, Jiu-Jitsu, and the other warrior training undergone by samurai (kendo, kyudo, iaido, etc.). All other forms of Karate came from Okinawan gentry.
At the root of it all, all karate stems from the same place -- style names evolved not out of differences in styles (initially -- they are very different now, but weren't at the beginning), but from names given to or given by the instructors. To use Shotokan as an example again: Shotokan was Gichin Funakoshi's pen-name from his short-lived days as a poet and literally means "Wind blowing through pine trees" (or something roughly equivalent). When he opened his own dojo and started teaching his own classes, his students posted the name "Shotokan" above the door as a sign of respect for their teacher, whom they loved so much.
Similarly, Shudokan simply means "the hall for the study of the (karate) way." Again, the difference in style vs. other types of karate is simply a process of evolution throughout the years of teaching different students and then THEY teach their students slightly differently, etc. etc. etc.
Also, Kung-Fu should not really be used to describe any Chinese fighting style -- it's a misnomer that has been terribly abused. Kung-Fu literally means a high level of skill -- you can be a kung-fu painter, a kung-fu poet, a kung-fu actor... I think the term you're looking for is Wushu.
But you're right, Kempo does differ from Karate, and was also known as Chuan-Fa, which influenced early Karate (or To-Te, or Okinawa-Te, or simply Te -- which are all names that contemporary Karate have used in the past), but they are not considered the same.
But to get back to the topic at hand -- Karate and what it is... Hard to answer in simple terms, as it means something different to everybody. To truly answer this question, you should ask YOURSELF "what is Karate to ME?" I guarantee, that's the best way to get the best answer.
