I'd expect mass takedowns of all sorts of things as soon as the legislation passes. Possibly a full shutdown of YouTube, megaUpload, and other sites deemed to be "encouraging piracy", but I wouldn't offer strong odds on that.
SOPA will become US law and either Youtube or Megaupload will be shutdown.
Additionally, you can expect other nations to get sick of random things being pulled from the DNS registries (basically huge lookup tables that tell your computer what to do with things like "www.lesswrong.com"; changing these to a "this page has been removed" page is the SOPA method-of-choice for taking a site down). An independent DNS network will probably be created, segmenting the internet into a US internet and a global internet.
So I know we're not supposed to get into this on LW (politics, minds, death, etc), but I figure opposing the censorship of the internet is alright to do.
Scream loudly at your local representatives (if you are a US citizen), and encourage others, particularly the young voters who are very attached to the internet, to do the same. If you're not a citizen, try writing someone in a diplomatic position to protest the censoring of a global resource for the benefit of US corporate interests (everyone uses the same DNS servers at the moment, so you'll be seeing the results whether you live here or not).
You are so mind-killed that you non-jokingly advocated folk activism on Less Wrong as the first method to stop the "censorship of the internet". Step away from the keyboard, take some deep breaths, multiply some probabilities, and come back after you've calmed down.
Interestingly, your link is to a political think-tank site. (Of course, argument screens off authority and all that---it looks like a pretty good article.)
Referring, of course, to the proposed U.S. legislation which could cause severe damage to the Internet—at least, that's what a lot of people are saying. See, e.g., this Open Letter From Internet Engineers to the U.S. Congress (the first signatory listed is Vint Cerf). On Wikipedia, people including Jimbo Wales are discussing strategies as extreme as blanking the entire site (except for an explanatory message) to get people's attention, and thereby perhaps incite them to action, such as calling their Congressional representative.
I just happened to find out about all this a few hours ago, being someone who tries to avoid distractions like most kinds of news, so possibly others here with similar habits will appreciate having it called to ther attention. Or possibly they won't. But to those of you who possess relevant kinds of expertise:
(I think this subject can be discussed without political advocacy, in which I am mostly not at all interested anyway. It just looks like a practical problem to me.)
Edited to Add: I forgot to include a fourth bullet point:
It seems to have been assumed by many commenters, nevertheless.