How to determine why the fan is running on MacBook Air?

The fan is turning on because the CPU is being taxed. You can look at what processes are using the CPU with Activity Monitor:

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Activity Monitor is a built-in application. Type "Activity Monitor" into Spotlight and it should be the first result.

Activity Monitor defaulted to show only "My Processes". None of those seemed to be the source of the problem.

However, when I switched the Activity Monitor view to "All Processes" Carbonite was the clear culprit. About 60 seconds after I set Carbonite to "Pause for 24 hours" its CPU utilization dropped to < 1% and my annoyingly loud fan slowed down then stopped.

I'm only using Carbonite, because I had a few months left on my old license, but I'm not sure its worth listening to a noisy fan for hours on end while it trickles my bits to the cloud.