California Style refers to the West Coast pizza styles that morphed out of upscale LA pizzarias in the mid-late 1970's. The guy who started the California Pizza chain started out at a small place where he (just a pizza maker) asked the owner if he would put some of his own creations on the menu. Not really thin but medium-thin crust pizza with more authentic parmesian and then toppings like shrimp, calamari, pineapple, etc and then a very light layer of mozz. Word got around about the "different" place and celebs/movie stars/tv stars, etc., started showing up and the guy got famous. He broke away and started his own place and the chain did well...even expanded out to Las Vegas (inside a major casino) and now has frozen versions nationwide. The guy went to Italy/Sicily for a few months and came back and expanded the menu to include totally authentic pies that are very different from what people who grew up in the Pizza Hut Generation expect.
Chicago Deep Dish is my personal favorite, some New York pies are runner up.
If you have a "Godfather's" in your area feel LUCKY. In fact, feel VERY LUCKY. They have really stayed loyal to their original plan: have the best made to order pies possible and put on the best salad bar around and do it for a decent price. For a chain, they are about the best.