The "war," which is described at another site under the name "Project Chanology" (a reference to 4chan, a popular message board, where most posts are made by people who don't login and are thus attributed to "Anonymous"), calls for denial of service attacks over the Internet, prank phone calls, spam emails, and personal visits involving vandalism and harassment. Apparently Scientology's main website was down due to denial of service for at least part of the day yesterday.
The press release cites a number of web pages for further information about Scientology, the second of which is the article "Scientology v. the Internet: Free Speech & Copyright Infringement on the Information Super-Highway" which Jeff Jacobsen and I wrote for Skeptic magazine in 1995 after Scientology effectively declared war on the Internet. (A much lesser-known sequel to that article, published only on the web, is "Scientology v. the Internet: An Update and Response to Leisa Goodman.")
I completely disagree with the tactics being used here--Scientology has as much right to free speech and protection of their copyrights as anyone else, though I also condemn Scientology's habitual misuse of copyright to try to suppress fair use of information. To the extent this is a prank designed to get media attention, well done. To the extent it gets taken seriously, though, it's something that may not end well. Read the material, watch the videos, have a laugh, and tell others about the absurdity and abuses of Scientology. But please, don't launch attacks on their websites, harass individuals, or engage in vandalism.
"Anonymous" previously received coverage for attacks on MySpace accounts on Fox 11 in Los Angeles on July 26, 2007.
BTW, the press release gets its facts wrong when it claims that the alt.religion.scientology Usenet newsgroup was "shut down." Scientology attorney Helena Kobrin issued an rmgroup message, but almost all news servers ignored it. The accurate facts may be found in Jeff's and my Skeptic article.
UPDATE: Wikinews and Xenu.net have more.