The preposition between is from an Old English word related to the word two. It denotes a position between two things:
The guard stood between the door and the street.
Just between you and me, I’m surprised that a graduate of Yale wouldn’t speak better English.
The preposition among derives from an Old English word meaning to mix or to mingle.
Use among when speaking about more than two things:
We wandered among the poppies, looking for the road to Oz.
Let’s keep this information among ourselves.
A common error is to use between where among is more appropriate:
I was one of eight brothers. Our parents never made any difference between us.
Better: Our parents never made any difference among us.
Here are some quotations from newspapers:
… storefronts in downtown Robersonville in North Carolina, one of 26 states where deaths now outnumber births among white people. … (www.nytimes.com)
Condom use is declining among sexually active teens, a federal survey found. … (www.nytimes.com)
sounded the alarm about the use of benzodiazepines among older adults. Often called “benzos,” the problem drugs include Valium (diazepam), Klonopin (clonazepam), … (www.nytimes.com)
SEOUL, South Korea – A top lieutenant of North Korean leader Kim Jong Un is in the United States conducting one of three sets of parallel talks aimed at salvaging a summit between Kim and President Donald Trump. usatoday.com