What is purpose of these arm band? In anime, people who in charge sometimes wear armband,



from left to right: Haruhi Suzumiya - The Melancholy of Haruhi Suzumiya (Club president), Suzu and Tsuda - Seitokai Yakuindomo (student council), Kuroko - Toaru Kagaku no Railgun (Judgement, Disciplinary Committee), Kamado Ueshita - Mirai Nikki (Director of the orphanage).

is this arm band have particular name? what is it purpose and how is this originate? Is this related to Japanese culture? I never see someone wearing this in my country.
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.


Source

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.