http://shogun.bitcrumbs.com/
Provinces are those different colour shapes on the map. Most provinces have different specialties which offer different bonus to the player/AI who holds the province. And yes, you need to take the castle/fortification for you to build and have those bonus. Farms in the more fertile provinces offer more income than those in less fertile ground, but the food offered by the farms are the same regardless of where they are built.
Hope that answers your questions.
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