What are Raster and Vector data in GIS and when to use?

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What are raster and vector data in the GIS context? In general terms what applications, processes, or analysis are each suited for? (And not suited for!)

In answering someone's question today, I wanted to point to a resource giving a general overview of these primary data representations, and to my surprise I couldn't find something like this overview written with GIS in mind. Let's fix that.

Update: Anyone have some small, concise, effective pictures which convey and contrast these two fundamental data representations? The answers so far are great, technical, correct, and wordy. For people new to the idea of raster and vector it will be easier to absorb the details if they have images to reference them to.



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