Arm bands in Japan, usually called as red armband, is worn as a method of identification of team/group or organization (more on this at the end)
The red armband can have a very different connotation in Japanese
culture than in that of many other countries. The leader of any group
might wear a red armband, usually with the characters for the name of
the organisation written on it, as a symbol of his authority. Due to
cultural diffusion, the usage of the red armband by leftist political
groups has spread to Japan as well.
The link also has information on how it spread to Japan but that's out of the scope so I'm not including it here.
Known bands from the picture:
1) Haruhi Suzumiya wears a band that reads Brigade leader
2) In Ousai (Private Academy featured in Seitokai Yakuindomo) red armbands signifies power, not
leadership—students that enforces rules wear them.
3) In Toaru Kagaku no Railgun, this band signifies the Judgement Committee
Couldn't find information regarding the fourth.
Also, personally, in many Japanese Dramas I have seen working people wear such bands like the reporters, journalists, forensics and even the police officers (not red though, it's usually white in black text) so I'm guessing it's a way to tell who's from which organization.