"The Internet interprets censorship as damage and routes around it." - Internet pioneer John Gilmore
Back in 2003, aerial photographer Kenneth Adelman photographed hundreds of miles of California coastline as part of a government-sanctioned effort to document coastal erosion. Of the 12,000 photographs he took and posted online, one happened to include an opulent cliffside mansion belonging to none other than Barbara Streisand. She sued, citing privacy concerns. Not only was her suit dismissed, but the picture of her home went viral, and suddenly what had been an extremely obscure part of a giant project hidden deep within the Internet was featured on blogs everywhere, ultimately being viewed a half-million times. Thus was coined the "Streisand Effect."
There are lots of examples of the SE in action. A pre-Internet example is banned book month, celebrating and highlighting literature that the powers that be have tried to censor. More recently, the Wikileaks website was the targeted for takedown by government agencies; soon after, people sympathetic to their cause mirrored Wikileaks' site across the world, making it impossible to completely remove. (It's kind of like trying to kill a worm by chopping it in half -- then you've got two worms.)
Here's a crazy one: in 2009, Ted Alvin Klaudt, a former South Dakota state legislator convicted of raping his two foster daughters, attempted to claim that his name was "copyrighted" and demanded it not appear in any news articles. This didn't work, of course, and his ridiculous claim got him lots of new publicity.
A recent and relevant example from world politics: in 2007, Tunisia blocked access to Youtube and DailyMotion after a video of Tunisian political prisoners was posted. Activists and their supporters then started to link videos about civil liberties in general, as well as the exact location of the Tunisian President's palace on Google Earth. The Economist wrote that this "turned a low-key human-rights story into a fashionable global campaign." Another example from an African dictatorship: Anonymous vs. the government of Zimbabwe. After dictator Robert Mugabe's wife Grace sued a newspaper for $15 million for publishing a Wikileaks cable linking her and some cronies to the illegal mining and fencing of blood diamonds, the "hacktivist" group Anonymous attacked government websites, even defacing the Ministry of Finance site to read thusly (see above).
Wow. Streisand in full effect.