The arm-bands are miniature playbooks. They have the name of certain plays and the specific instructions for the player's position. This keeps the players from having to memorize hundreds upon hundreds of plays, and gives them a cheat-sheet if they forget something.
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Exactly what this poster said, it's only a fashion statement, the amount of compression you would need can't be provided by a little rubber strip. If you notice they're almost always Nike or Under Armor, so it's another place to advertise really.
I disagree with the two previous posters. I believe its similar to soccer players wrapping just below the knee. It stabilizes the tendon running down the front of your knee, and prevents pain from running. It does not actually take much pressure to do this, which most people do not realize.
The band above the arm (when its a type of wrap, not a sweat band) can help prevent stretching ligaments in your elbow that are easily injured when you hit the ground.
Some believe that the elbow straps are purely cosmetic and have no true purpose, but it is hard to believe that the NFL would allow its players to wear any on field equipment with no purpose. I lean towards the believe that the straps help with tendinitis of the elbow.