Anonymous is a hacktivist group that has been around since about 2004. A hacktivist is somebody who is politically protesting over the internet. The group originated from the website 4chan and has slowly evolved since then. Their first big protest was against the Church of Scientology where members across the world where told to meet at the nearest church wearing Guy Fawkes Masks.
Most recently Anonymous has participated in assisting with the revolution in Egypt after the previous dictator had shut down the internet to the entire country. They have also played a strong roll supporting the Occupy Wall Street movement.
Other events they have protested against include; The Pirate Bay, Megaupload, SOPA (Stop Online Piracy act), Kony, Government Websites, Internet Pedophilia, and other religious organizations.
In 2012 Times Magazine named Anonymous as one of the most influential groups of people in the world.
To watch a documentary on the group feel free to go here.