"Anonymous" consists of a collection of people who act based upon conversations on the marginally moderated 4chan.org /b/ forum. /b/ is the site's "Random" board. It usually features pornography and efforts to get 4chan readers to harass a particular individual. Most of the time these efforts consist of "I hate this bitch, here's her facebook / home number, make her suffer!" These are usually met with the reply of "Anonymous is not your personal army" and a collective "meh."
If, however, kittens are being tortured, rabbits crushed, puppies thrown into rivers or the Internet censored, 4chan declares war.
A few key hackers will post download links to botnet clients to turn your PC into a zombie for a DDoS attack on targeted web sites. At various points over the last few days, Mastercard, Paypal and Visa have all been unable to process any transactions due to these attacks.
The better researchers will dig up loads of information to be used in tracking down the offenders. This, for example, is how the woman who tossed a cat in a trash can in the UK was tracked down.
The war will escalate until, I kid you not, the warriors grow bored or move onto something else. Frequently this happens after a sufficiently large laugh comes out of the exchange. For example, an 11 year old girl was posting topless photos of herself to /b/ and links to videos where she talked about using a gun to make a "Brain Slushie" out of anyone who gave her grief. Anonymous tracked down her home phone and made it very clear to her parents what kind of person she was being. They were not kind. They were raging assholes. The issue died when her father, displaying a complete disregard for the fact that his 11 year old started this by posting nudie photos of herself online, posted a video where he threatened 4chan. Normally this would only escalate things, but he made such an idiot of himself Anonymous just laughed at him, treating his "Consequences will never be the same (NSFW!)" video as the best possible punch line to the entire affair.
If you had to translate Anonymous into D&D terms, the best match for their alignment would be "Chaotic asshole."
Side note, most of the porn on /b/ falls into one of the following categories:
1. Chubby porn.
2. Female 4chan readers posting their own topless photos.
3. Shemale porn.
4. "Trap" porn, referring to young effeminate men who can pass for women when properly tarted up.
5. "Dumps" where someone starts dumping their favorite porn photos, most of which were found on 4han to begin with.
Every few hours some idiot tries to start a child pornography thread. The moderators step in quickly, lock the thread, delete it and allegedly turn the IP addresses of the idiots who posted the garbage over to the FBI.
4chan is hosted on the USA. Its founder, Moot, is a US citizen. No one really knows HOW the site has avoided being taken down by the US government. The best guess is that that FBI finds it too rich a source of information on child porn producers and traders.
It's safe to say that none of the conservative pundits have ever looked at 4chan themselves. If they had, they'd be decrying /b/ as part of the gay agenda. You can't go more than a few pages on /b/ before hitting a tranny thread. It can be quite jarring until you become desensitized to the sight of a 6 foot tall ebony black transsexual with implants felating herself.
Then again, given the Republican track record of the last few years, they may know EXACTLY what's on /b/ and are keeping it quiet so as to not dry up one of their favorite sources of spank bank imagery.