The anonymous are a decentrilized leaderless group focused on cyber-activism and hacking. Their symbol is the guy fawkes mask from the movie V for vendetta. They where founded in around 2004 as an online community from 4chan's /b/ board they where focused on entertainment but starting from 2008 with the hacking of the Church of Scientology they became more of a group focused on hacking. They went on to hack or bring offline services from companies and organisations including the Westboro Baptist Church, PayPal, MasterCard, Visa, and Sony. Their members range from script kiddies to exploit coders but most of them are teenagers and neophytes in the hacking scene.
The motives depend on members, most do it for political motives some for glory and some just for the "lolz". But the group in general have risen up against goverments and laws they deemed breaching peoples liberteys, they attacked the tunisian goverment during the 2011 arab spring they protested against ACTA and have hacked child pornography sites. In 2013 they launched a campaign against Israeli website because they say they are against Zionism.
I do not like the Anonymous most of Their members are incompetent. And Their attacks are based on a lot of their members using the program LOIC to DDoS or on the few skilled ones. Mush smaller groups like LulzSec or UGNazi have caused much more havoc, which in the end is what they want to cause, even if they hide their hacking behind "political motives".