Use between when you are thinking of two things at a time, even though they may be part of a group consisting of a larger number.
I loved to sit between my two brothers in the back seat of the car on family road trips.
The team certainly needed that short rest period between quarters. (Although there are more than two quarters, a rest period occurs only between two of them at a time).
Use among when you are thinking of a group of three or more rather than of separate individuals.
Although there was a significant disagreement among the officers about the theme for Sports Night, we eventually compromised.
Since we had only $25 cash among the six of us, I decided to pay with my credit card.