I have a location's latitude and longitude as well as the complete address. How do I figure out if it is an urban area or a rural area? Is there an API service available or something else I could use?
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You'll probably need commercial data for that. Rural Urban Suburban Codes is a general designation given by government agencies for each zip code (often used for determining grants and tax incentives - and some agencies define it differently). You can learn a more from the link: Rural Urban Suburban (don’t miss the detail tab at top)

puddleglum
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Barry is completely wrong. This information is not subjective. It is available from the US Census Bureau. You can also search on Rural Urban Continuum Codes.
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Interesting, although it doesn't seem to be global data. – barry Dec 03 '18 at 08:24
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I doubt whether this type of information is available as one man's rural could be another man's urban. It's subjective.

barry
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https://gis-portal.data.census.gov/arcgis/apps/MapSeries/index.html?appid=7a41374f6b03456e9d138cb014711e01 – Abram Nov 30 '18 at 16:27