According to the New York Times article (see related links below), the armbands are nothing more than the current fashion trend with the purpose of showing off their upper arm definition.
Wearing the armbands (in many cases wristbands moved up to the upper arm) has no medical or performance benefit.
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The black bands could be used for grip of the ball by the running backs, or some players wear them to reduce swelling in the elbows.
They are worn in mourning, as a sign of respect for a deceased person that the players knew.