What does the “coursing” mean in this sentence？
"Her pulses beat fast, and the coursing blood warmed and relaxed every inch of her body."

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The word means 'flowing' as Rory said. It is a word used also for the flow of a river.
For example:
"Be a man
We must be swift as the coursing river"

which is from this song: https://www.youtube.com/watch?v=TVcLIfSC4OE

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The sentence is from a very interesting story I read yesterday. “The story of an hour” written by Kate Chopin.

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Flowing through her body. She's excited and the blood is flowing fast around her body, really romantic novel stuff, hope it ended well!! ;-)