I’ve noticed these armbands on NFL players in the last year or so (See this picture of Randy Moss for an example.) They are tight and seem to be worn just above the elbow / at the bottom of the bicep. What are they called, and what purpose do they serve?
Sweat bands?
They keep the arms from falling off of the robots. Ok, seriously, they are sweatbands hiked up above their elbows. Probably for dramatic effect/machisimo more than actual utility.
I agree with that pretty much.
They may serve some purpose when the ball is caught and tucked for running.
According to Nike’s online store, these are sold as “bicep bands” ($5):
“Flash your guns during the game with these sweat-defying bicep bands.
Cotton and nylon blend with added spandex ensures super performance and a great fit.
Embroidered Swoosh adds sporty flair.”
They must not help your catching ability at all if Randy Moss is sporting them…
I think the first sentence there pretty much tells it all.
There can be a reason for sweat bands there, though. When I run and it’s warm out (especially if I’m trying to push myself a bit), I’ll suddenly have sweat dripping down to my elbow. Funny thing is that it’s not coming from my armpits. 
But bands like that would be useful, since I find trickling sweat on my arms to be ticklish and annoying.
Like all arm/head bands they are decorative. I wore a big band on my right forearm in High School because Michael Jordan wore one on his left arm. He was right handed I am left so we wore them on our off hand. My son started wearing one of those small bands on his bicep because well all the other kids do.
they help cover up their needle trac marks 
Um, there’s a difference between a pro football player in a game, sweating his ass off… and just wearing one for the fun of it.
Sweatbands do serve a purpose. This is wild, I know, but they: absorb sweat. I love my head one because it keeps sweat from going into my eyes, which stings really badly. Wrist ones are useful for wiping an upper lip or any possible spit/snot that might occur. And like I said in the post immediately above yours, I’d find those bicep ones pretty darn useful.
I met a guy on a group bike ride that wore a piece of rubber or plastic tubing on his head. It didn’t absorb the sweat, but it kept the sweat out of his eyes and guttered it around back.